LIBRARY 

OF  THE 

UNIVERSITY  OF  CALIFORNIA. 


Class 


WORKS  OF  PROF.  J.  L.  R.  MORGAN 

PUBLISHED    BY 

JOHN  WILEY  &  SONS. 


The  Elements  of  Physical  Chemistry. 

Third  Edition,  Revised  and  Enlarged,  izmo, 
xii  +  sio  pages.  Cloth,  $3  oo. 

Physical  Chemistry  for  Electrical  Engineers. 

i2mo,  viii  +  230  pages.    Cloth,  $1.50  net. 

TRANSLATION. 
The  Principles  of  Mathematical  Chemistry. 

The  Energetics  of  Chemical  Phenomena.  By  Dr. 
Georg  Helm,  Professor  in  the  Royal  Technical 
High  School,  Dresden.  Authorized  Translation 
from  the  German  by  J.  Livingston  R.  Morgan, 
A.M.,  Ph.D.,  Adjunct  Professor  of  Physical 
Chemistry,  Columbia  University.  lamo,  viii  +  228 
pages,  cloth,  $1.50. 


PHYSICAL  CHEMISTRY 


FOR 


ELECTRICAL  ENGINEERS 


BY 


J.    LIVINGSTON    R.    MORGAN,    PH.D. 

Professor  of  Physical  Chtmistry  in  Columbia  University 


FIRST    EDITION 
FIRST    THOUSAND 


NEW  YORK 

JOHN   WILEY   &    SONS 

LONDON:    CHAPMAN  &   HALL,    LTD. 

1906 


Copyright,  1906 

BY 
J    L.  R.   MORGAN 


ROBERT  DRUMMOND,   PRINTER,   NEW  YORK 


PREFACE. 


THIS  book  is  intended  not  only  for  the  professional 
electrical  engineer,  but  also  for  the  use  of  all  those  who 
have  the  same  object  in  view,  viz.,  the  attainment  of  a 
knowledge  of  physical  chemistry  sufficient  in  its  scope 
for  the  understanding  of  current  work  in  electrochemistry. 

Although  electrochemistry  lies  in  that  border-land 
between  chemistry  and  electricity,  it  has  been  so  con- 
sistently neglected  by  the  majority  of  the  workers  in  both 
fields,  that  its  development  has  rested  almost  entirely  in 
the  hands  of  a  comparatively  small  number  of  specialists. 
The  time  has  now  arrived,  however,  when  the  results 
obtained  by  these  men  are  generally  recognized  as  of 
exceeding  importance,  no  less  from  the  practical  than 
the  theoretical  standpoint,  and  consequently  a  working 
knowledge  of  the  subject  has  become  a  very  essential 
part  of  the  equipment  of  the  student  who  is  to  devote 
himself  to  either  branch  of  science. 

But  even  a  glance  on  the  part  of  such  a  student  at  one 
of  the  electrochemical  contributions  which  at  present 
fill  so  many  pages  of  our  journals  is  sufficient  to  make 
clear  the  fact  that  the  entire  work  is  based  upon  an 
utterly  new  and  unfamiliar  set  of  laws  and  concepts;  in 
short,  upon  a  knowledge  of  certain  portions  of  physical 

iii 


IV  PREFACE. 

chemistry,  and  that  without  this  knowledge  any  at- 
tempt to  understand  the  attaining  of  the  result,  or  even 
its  meaning  when  attained,  is  futile. 

The  purpose  of  this  book  is  to  aid  those  who  find  them- 
selves in  such  a  position,  and  to  present,  in  as  popular 
a  form  as  is  consistent  with  quantitative  results,  those 
laws  and  generalizations  of  physical  chemistry  which 
form  the  basis  of  the  subject  embracing  the  chemical 
application  of  electricity  and  the  electrical  applica- 
tion of  chemistry.  It  is  in  nowise  to  be  regarded  as 
a  text-book  of  electrochemistry,  however,  for  although 
that  subject  is  discussed  in  considerable  detail,  only  those 
portions  of  it  are  considered  which  best  illustrate  the 
application,  and  methods  of  application,  of  the  physical 
chemical  principles  already  presented,  and  no  com- 
plete account  is  attempted.  But  notwithstanding  this 
the  reader  should  gain  a  very  clear  and  just  idea  of  the 
elements  of  electrochemistry,  and  one  which  he  can 
readily  elaborate  by  further  work. 

While  a  cursory  glance  might  easily  lead  one  to  infer 
that  some  portions  which  are  treated  at  length  are  utterly 
unnecessary  for  the  electrochemist,  such  is  not  the  case, 
for  everything  presented,  if  not  vitally  important  in 
electrochemistry  itself,  is  absolutely  requisite  for  the 
understanding  of  something  else,  which  is  absolutely 
indispensable  to  it.  The  reader  is  cautioned,  therefore, 
not  to  omit  anything,  thinking  it  unessential  or  trivial, 
for  everything  is  given  with  a  definite  object  in  view. 

Since  the  subject  of  solution  is  also  the  most  important 
portion  of  physical  chemistry  for  those  specializing  in 
other  branches  of  chemistry,  the  book  is  also  adapted 
to  the  use  of  chemists  who  have  but  a  limited  time  to 
devote  to  the  subject,  or  whose  need  is  restricted  to  a 


PREFACE.  V 

knowledge  of  the  behavior  of  substances  in  solution. 
And  it  may  also  serve  as  an  introduction  to  a  more  com- 
plete course  in  physical  chemistry,  for  the  numerous 
references,  both  to  originals  and  to  a  more  elaborate 
work,*  will  enable  the  student  to  readily  investigate 
further  any  point  upon  which  his  interest  may  happen 
to  be  aroused. 

In  order  that  the  reader  may  gain  a  working  idea  of 
the  subject,  a  collection  of  problems,  together  with 
their  answers,  is  given  in  the  final  chapter.  The  solution 
of  these  will  not  only  make  the  principles  of  the  subject 
more  real  and  clear,  but  will  serve  to  impress  more 
deeply  upon  the  reader's  mind  the  essential  and  gener- 
ally useful  portions  of  the  various  laws. 

But  one  other  point  need  be  mentioned  here. 
Throughout  the  presentation  /  have  avoided  the  use  oj  any 
hypothesis,  feeling  that,  by  placing  the  subject  upon  an 
absolutely  experimental  basis,  giving  a  practical  experi- 
mental definition  of  each  concept  as  it  is  used,  and  draw- 
ing no  inference  not  justified  in  all  its  parts  by  actual 
results,  the  reader's  idea  will  be  the  more  clear  and 
scientific. 

J.  L.  R.  M. 

COLUMBIA  UNIVERSITY,  September,  1905. 

*  Morgan,  The- Elements  of  Physical  Chemistry,  30!  edition,  1905. 


CONTENTS. 


CHAPTER  I. 

PAGB 

SOME  FUNDAMENTAL  PRINCIPLES i 

Atomic  and  molecular  weights,  I.  Energy,  u.  The  factors 
of  energy,  13. 

CHAPTER  H. 

THE  GENERAL  PROPERTIES  OF  GASES 16 

The  gas  laws,  16.  Dissociation,  23.  Partial  pressures  and 
concentrations,  30. 

CHAPTER  III. 

HEAT  AND  ITS  TRANSFORMATION  INTO  OTHER  FORMS  OF  ENERGY  34 

The  first  law  of  thermodynamics,  34.  The  second  law  of 
thermodynamics,  44. 

CHAPTER  IV. 

SOLUTIONS 50 

The  formula  (molecular)  weight  in  the  liquid  and  solid 
states,  50.  Osmotic  pressure,  54.  Vapor  pressure,  68.  Boil- 
ing-point, 70.  Freezing-point,  72.  Coefficient  of  distribu- 
tion, 74.  Electrolytic  dissociation  or  ionization,  76.  The 
thermal  relations  of  electrolytes,  90. 

CHAPTER  V. 

CHEMICAL  MECHANICS. 97 

The  law  of  mass  action,  97.  Equilibrium  in  gaseous  sys- 
tems, IQI.  Equilibrium  in  liquid  systems,  114.  The  effect 

vii 


viu  CONTENTS. 

PAGE 

of  temperature  upon  the  equilibrium-constant,  118.  Velocity 
of  a  chemical  reaction,  124.  Reactions  of  the  first  order. 
Catalysis,  128.  Reactions  of  the  second  order,  130. 

CHAPTER  VI. 

EQUILIBRIUM  IN  ELECTROLYTES 133 

Organic  acids  and  bases.  The  Ostwald  dilution  law,  133. 
Acids,  bases,  and  salts  which  are  ionized  to  a  considerable 
extent.  Empirical  dilution  laws,  140.  The  heat  of  ionization, 
145.  Solubility  or  ionic  product,  148.  Hydroly tic  dissociation, 
or  hydrolysis,  157.  Ionic  equilibria,  168.  The  color  of  solu- 
tions, 170. 

CHAPTER  VII. 
ELECTROCHEMISTRY. 172 

The  migration  of  ionized  matter. 

Faraway's  law.  Electrical  units,  172.  The  migration  of 
ionized  matter,  174. 

The  conductivity  of  electrolytes. 

The  specific,  molar,  and  equivalent  conductivities,  180. 
Ionic  conductivities,  181.  Empirical  relations,  187.  The 
ionization  of  water,  188.  The  solubility  of  difficultly  soluble 
salts,  189. 

Electromotive  force. 

The  chemical  or  thermodynamical  theory  of  the  cell,  191. 
The  osmotic  theory  of  the  cell,  193.  Differences  of  potential. 
Calculation  of  the  electrolytic  solution  pressure,  202.  The 
heat  of  ionization,  204.  Concentration  cells,  205.  Determina- 
tion of  ionization  from  electromotive-force  measurements,  209. 
The  processes  taking  place  in  the  cells  in  common  use,  211. 

Electrolysis  and  polarization. 

Decomposition  values,  214.  Primary  and  secondary  decom- 
position of  water,  218. 

CHAPTER  VIII. 
PROBLEMS > 220 


OF  THE 

(    UN1VER- 


PHYSICAL  CHEMISTRY  FOR  ELECTRICAL 
ENGINEERS. 


CHAPTER  I. 

SOME  FUNDAMENTAL  PRINCIPLES. 

Atomic  and  molecular  weights. — From  the  very  defi- 
nition of  physical  chemistry — that  portion  of  science 
which  has  for  its  object  the  study  and  investigation  of 
the  laws- governing  chemical  phenomena — it  is  apparent 
that,  we  have  to  do  with  the  most  general  and  inclusive 
of  all  the  branches  of  chemistry.  Certain  of  its  funda- 
mental concepts,  therefore,  are  those  which  are  also 
fundamental  to  all  these  branches,  and  it  is  essential 
that  we  at  least  recall  them  to  mind  before  proceeding 
to  the  consideration  of  the  things  based  upon  them. 

Two  of  the  concepts  common  to  all  branches  of  chem- 
istry, the  atomic  weight  and  the  molecular  weight, 
play  an  especially  important  r61e  in  physical  chemistry, 
as  they  do,  indeed,  in  all  the  other  branches  of  chemistry. 
It  is  unfortunate,  however,  that  the  student's  usual  idea 
of  these  is  such  an  utter  confusion  of  fact  and  hypothesis 
that  no  very  clear  conception  of  their  meaning  is  pos- 
sible. This  is  particularly  evident  at  the  present  time, 


2     PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

when  it  is  by  no  means  an  uncommon  thing  to  find  those 
who  actually  fear  for  the  security  of  the  quantitative, 
experimental  results  of  chemistry,  because  of  the  pos- 
sibility of  the  so-called  atom  proving  divisible;  in  other 
words,  making  the  definition  of  the  hypothetical  atom, 
and  hence  also  that  of  atomic  weight,  a  meaningless  col- 
lection of  words. 

Without  in  any  way  condemning  these  hypotheses 
more  than  the  others  in  common  use,  it  must  be  ad- 
mitted as  far  better  to  recognize  that  our  science  is  based 
upon  no  such  insecure  and  uncertain  foundation,  but 
upon  facts  which  will  continue  to  exist  under  any 
hypothesis,  or  lack  of  hypothesis,  and  which  at  any  time 
can  be  confirmed  by  experiment.  That  such  is  the 
foundation  of  chemistry  is  undeniably  true,  but,  un- 
fortunately, the  lay  mind,  and,  still  more  unfortunately, 
the  average  professional  mind,  refuses  utterly  to  recog- 
nize and  acknowledge  it. 

As  in  our  present  work  we  are  to  depend  solely  upon 
fact,  avoiding  all  hypothesis,  it  is  quite  essential  that  we 
secure  from  the  very  beginning  a  hypothesis-free  idea 
of  atomic  and  molecular  weight.  The  first  step  toward 
this  end,  however,  is  to  define  exactly  what  we  mean  by 
the  word  hypothesis,  for  common  usage  indicates  that 
its  significance  is  not  uniform.  A  hypothesis,  in  the 
sense  we  are  to  use  it,  is  a  theory  into  which  is  incor- 
porated something  that  is  foreign  to  the  facts  observed; 
the  word  theory,  as  we  shall  continue  to  employ  it, 
signifying  a  generalization  of  observed  facts,  containing 
nothing  beyond  what  is  expressed  by  these  facts.  A 
theory,  then,  is  a  law  of  nature  holding  between  certain 
well-defined,  if  narrow,  limits,  which  becomes  a  general 
law  when  the  limits  have  been  sufficiently  extended.  A 


SOME  FUNDAMENTAL  PRINCIPLES.  3 

mathematical  formula,  thus,  provided  each  term  is  de- 
terminable  by  experiment,  is  no  hypothesis,  but  a  theory 
or  a  general  law,  according  as  its  limits  are  restricted  or 
wide.  A  hypothesis,  on  the  other  hand,  when  in  the. 
form  of  an  equation,  contains  a  term  or  terms  which 
cannot  be  determined  by  direct  experiment;  and  in  any 
other  form  contains  assumptions  foreign  to  the  facts 
observed,  assumptions  which  sooner  or  later  must  be 
found  to  be  in  disagreement  with  fact.* 

In  our  work,  therefore,  we  must  test  each  concept  as 
it  arises  in  order  to  see  that  it  is  free  from  hypothesis, 
and  to  be  assured  that  we  have  to  do  solely  with  facts. 
Later,  naturally,  these  facts,  when  supplemented  by 
others,  may  be  included  in  more  general  laws,  but  facts 
they  will  always  remain,  notwithstanding,  and  independ- 
ent of  any  hypothesis  which  at  that  time  may  happen 
to  be  in  vogue.  Our  progress  will  thus  be  continuous,  and 
it  will  never  become  necessary  to  halt  for  the  purpose  of 
inquiring  what  is  fact  and  what  hypothesis. 

At  worst,  in  this  way  (retaining  hypotheses),  one  may 
be  certain  that  the  greater  emphasis  is  laid  upon  the 
facts  themselves,  and  that  the  hypotheses  can  lead  to  no 
serious  confusion.  At  best,  on  the  other  hand  (relin- 
quishing all  hypotheses,  as  we  shall  do),  we  can  devote 
all  our  energy  to  the  facts  and  their  generalization,  and 
thus  avoid  wasting  the  time  and  effort  requisite  for 
the  making  of  unwarranted  assumptions,  which  the 
progress  of  science  must  inevitably  prove  to  be  utterly 
untenable. 

That  any  one  who  has  read  of  the  determination  of 

*  For  a  masterly  discussion  of  these  things  see  Ostwald,  Vorlesungen 
uber  Naturphilosophie,  pp.  202-227,  r9O2>  a  book  to  which  I  am  in- 
debted for  this  point  of  view. 


4     PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

an  atomic  weight  can  retain  the  idea  that  it  depends 
in  any  way  other  than  name  upon  hypothesis,  seems 
impossible,  and  yet  the  proof  of  the  non-existence  of 
the  hypothetical  atom  would  seriously  disturb  the  ideas 
of  many  such,  notwithstanding.  That  an  event  of  the 
sort  could  affect  the  actual  results  of  the  science,  and 
certainly  the  actual  results  constitute  the  science,  is  not 
to  be  imagined  for  an  instant,  however.  In  the  words 
of  Ostwald — the  stoichiometrical  laws  will  continue  to  be 
a  part  of  chemistry,  even  after  the  time  when  the  atom  is 
only  to  be  found  in  the  dust  of  the  libraries. 

The  atomic  weight  of  an  element,  as  we  determine  it, 
is  simply  the  weight  which  will  combine  with  16  units 
of  weight  of  oxygen,  or  some  small  multiple  or  sub- 
multiple  of  16;  or  which  will  combine  with  the  atomic 
weight  so  determined,  or  some  small  multiple  or  sub- 
multiple  of  it,  of  any  other  element.  In  each  case,  there- 
fore, the  atomic  weight  is  an  experimentally  determined 
combining  weight.  Which  combining  ratio  is  to  be  selected 
in  case  two  are  discovered  (CO  and  CO 2,  for  example), 
and  which  weight  of  oxygen  is  to  be  employed  as  the 
standard  for  any  one  element,  are  matters  that  are  de- 
pendent only  upon  other  facts,  i.e.,  those  relating  to  com- 
pound substances. 

An  atomic  weight,  therefore,  has  no  vital  connection 
with  the  atomic  hypothesis,  but  is  based  solely  upon 
the  experimental  law — that  elements  combine  with  one 
another  only  in  the  proportions  of  their  combining 
weights,  or  of  small  multiples  of  them.  Naturally,  if 
atoms  exist  (in  accordance  with  the  usual  definition) 
they  must  be  individually  related  in  weight  as  the  experi- 
mentally determined  combining  weights,  called  atomic 
weights;  but,  whether  atoms  exist  or  not,  these  com- 


SOME  FUNDAMENTAL  PRINCIPLES.  5 

bining  weights  must  continue  to  be  true,  for  they  are 
facts,  independent  of  time,  and  can  always  be  tested 
by  experiment.  A  moment's  thought,  indeed,  will  show 
that  the  connection  between  the  hypothesis  and  the  actual 
results  is  always  as  slight  as  this.  Whenever  we  make 
use  of  the  word  atom,  in  practical  work  (for  no  other 
use  is  warranted),  it  is  the  atomic  weight,  the  gram- 
atom,  the  experimentally  determined  combining  weight 
that  is  intended,  and  never  the  hypothetical  atom.  It 
is  only  the  name  atomic  weight,  then,  which  leads  to 
the  inference  that  the  atomic  hypothesis  is  fundamental 
to  chemistry.  To  avoid  all  possibility  of  such  a  mis- 
conception here,  we  shall  employ  throughout  the  book 
the  word  combining  weight,  meaning  by  it  that  combining 
weight  which  is  usually  designated  as  the  atomic  weight. 

Much  that  has  been  said  concerning  the  atomic  weight 
is  also  true  of  the  molecular  weight.  For  the  molecular 
weight,  as  we  determine  it  in  the  gaseous  state  (we  shall 
consider  the  other  states  of  aggregation  later),  is  the 
actually  observed  weight  of  gas  which  occupies  the  same 
volume  as  32  units  of  weight  of  oxygen  under  like  con- 
ditions of  temperature  and  pressure.  And  again  here, 
if  we  assume  the  gas  to  be  composed  of  ultimate  particles 
(molecules)  of  uniform  size,  the  weight  of  the  molecule  of 
one  substance  will  be  related  to  that  of  another  as  are 
the  molecular  weights.  But,  whether  molecules  exist 
or  not,  the  so-called  molecular  weights  are  experiment- 
ally observed  facts,  and  as  such  are  independent  of 
hypothesis. 

According  to  hypothesis  the  molecular  weight  of  a 
substance  is  equal  to  the  sum  of  the  atomic  weights  of 
the  constituents.  Speaking  solely  from  the  experi- 
mental standpoint,  the  molecular  weight  is  that  weight, 


6     PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

found  by  a  summation  of  the  respective  combining 
weights  (p.  5),  which  will  occupy  the  same  volume  as 
32  units  of  weight  of  oxygen.  As  the  various  symbols 
of  the  elements  represent  the  combining  weights  of  these 
elements,  and  the  number  of  combining  weights  (atoms) 
is  always  given  by  the  subnumerals,  the  molecular 
weight  thus  being  indicated  by  the  formula,  we  shall 
designate  throughout,  in  our  later  work,  the  so-called 
molecular  weight  as  the  formula  weight  of  the  substance. 
Any  method,  then,  for  the  determination  of  the  formula 
(excluding  the  analytical  methods,  for  they  enable  us 
only  to  ascertain  the  ratios  of  the  weights  of  the  elements 
combining)  will  offer  a  means  of  finding  the  molecular, 
i.e.  the  formula  weight.  And  in  place  of  the  expression, 
the  molecular  (formula)  weight  in  grams,  we  shall  employ 
the  abbreviation  suggested  by  Ostwald,  and  call  it  the 
mole.* 

A  glance  at  the  methods  of  using  the  conceptions  of 
atomic  and  molecular  weight  is  amply  sufficient  to  show 
that  it  is  upon  the  practical  results  that  everything  is 
based,  and  not  the  hypothesis.  According  to  hypothesis, 
the  equation 

H2  +  C12=2HC1 

means  that  i  molecule  (2  atoms)  of  hydrogen  and  i  mole- 
cule (2  atoms)  of  chlorine  unite  to  form  2  molecules  of 
hydrochloric  acid  gas.  This,  of  course,  may  be  true, 

*  Since  the  normal  solution  of  analytical  chemistry  contains  one  equiva- 
lent mole  per  liter,  we  shall  use  the  word  molar  when  speaking  of  the 
number  of  moles  per  liter.  In  certain  cases,  then,  hydrochloric  acid 
for  example,  the  two  expressions  will  be  identical,  while  in  others, 
sulphuric  acid,  for  instance,  the  molar  solution  will  contain  twice  as 
much  as  the  normal  one,  etc. 


SOME  FUNDAMENTAL  PRINCIPLES.  7 

and  it  may  not,  for  we  can  neither  prove  it  nor  yet  dis- 
prove it,  but,  whether  it  be  true  or  false,  the  fact  still 
remains  that  2  units  of  weight  (i.e.  2  combining  weights, 
or  i  formula  weight)  of  hydrogen  and  71  units  of  weight 
(i.e.  2  combining  weights,  or  i  formula  weight)  of  chlor- 
ine unite  to  form  73  units  of  weight  (i.e.  2  formula  or 
combining  weights)  of  hydrochloric  acid  gas.  And  this 
was  true  before  the  hypothesis  was  formulated,  and  can 
always  be  proven  to  be  true  at  any  time  in  the  future, 
independent  of  the  hypothesis.  In  the  latter  case  we 
have  simply  expressed  facts  which  have  been  observed, 
and  can  be  observed  at  any  time;  in  the  former,  we 
have  added  to  the  observed  facts  the  assumption  of  an 
atomic  structure  of  matter,  which  is  foreign  to  the  facts 
themselves,  and  which  cannot,  like  them,  be  proven 
by  experiment  to  be  true. 

That  is  all  very  simple  and  true,  the  reader  may  say, 
now  that  we  have  formed  the  conceptions  of  atomic 
weight  and  molecular  weight,  but  how  could  such  results 
have  been  attained  without  the  aid  of  hypothesis?  The 
answer  to  this  question  is  that  everything  which  has 
been  attained  is  only  the  result  of  experiment,  and  that 
such  conceptions,  similar  in  all  but  name,  can  be  obtained 
without  difficulty  directly  from  the  actually  observed 
relations.  Starting,  for  example,  with  the  law  of  com- 
bining weights — that  elements  combine  with  one  another 
only  in  the  proportions  of  their  combining  *  weights  or 
of  small  multiples  of  them — and  the  law  of  combining 
volumes — that  gaseous  elements  combine  in  simple 
relations  as  to  volume,  or  in  small  multiples  of  them, 
the  volume  of  the  gaseous  product  formed  standing  in 
simple  relation  to  the  volume  occupied  by  the  constitu- 

*  Used  here  in  its  general  sense,  not  as  on  pp.  5  and  6. 

V 


8      PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

ents  originally — it  is  quite  evident  that  the  weights  oj 
equal  volumes  oj  gases  are  related  as  their  combining 
weights,  or  as  small  multiples  of  them.  We  might  infer 
from  this,  then,  that  the  combining  (atomic)  weights 
could  be  so  selected  that  the  density  of  the  various  gases 
would  be  proportional  to  their  combining  weights.* 

Experiment,  however,  shows  this  assumption  to  be 
incorrect.  For  example,  it  has  been  found  that  2  volumes 
of  hydrogen  unite  with  i  volume  of  oxygen  to  form  2 
volumes  of  gaseous  water;  that  i  volume  of  chlorine 
with  i  volume  of  hydrogen  forms  2  volumes  of  hydro- 
chloric acid  gas,  and  that  i  volume  of  gaseous  phos- 
phorus with  6  volumes  of  hydrogen  produces  4  volumes 
of  phosphine.  And  the  assumption  that  the  unit  of 
volume  is  that  volume  which  is  occupied  by  the  com- 
bining (atomic)  weight  of  an  element  leads  to  the  result 
that  the  density  (i.e.  the  weight  of  unit  volume)  of  water 
vapor  is  but  one-half  the  sum  of  the  combining  (atomic) 
weights  of  its  constituents,  which  is  also  true  for  the 
gaseous  hydrochloric  acid,  while  the  density  of  phos- 
phine is  but  one-fourth  the  sum  of  the  combining  (atomic) 
weights  of  the  constituents. 

If  we  retain  the  combining  weights  as  they  have  been 
determined,  however,  and  assume  the  unit  of  volume 
to  be  that  occupied  by  a  small  number  of  combining 
(atomic)  weights,  where  the  number  is  dependent  upon 
the  element,  it  is  at  once  evident  that  a  general  result 
is  obtained.  And  this  assumption  is  perfectly  justi- 
fiable and  unconnected  with  hypothesis,  for,  although 

*  The  combining  weight  of  a  compound  being  identical,  naturally, 
with  its  formula  weight,  i.e.,  equal  to  the  sum  of  the  combining  (atomic) 
weights  of  its  constituents.  Thus  for  water  the  value  is  18,  i  e., 
0+H+H. 


SOME  FUNDAMENTAL  PRINCIPLES.  9 

the  combining  ratios  (both  for  volumes  and  weights) 
are  experimental  facts,  the  choice  of  the  units  for  their 
expression  is  perfectly  arbitrary,  as  is  the  choice  of  any 
other  unit,  and  purely  a  matter  of  convenience. 

In  the  case  of  hydrochloric  acid  gas,  then,  for  example, 
assuming  the  unit  of  volume  to  be  that  occupied  by 
2  units  of  weight  of  hydrogen,  we  obtain  the  following 
result:  i  volume  of  hydrogen  (2  units  of  weight)  will 
combine  with  i  volume  of  chlorine  (2X35.5  units  of 
weight)  to  form  2  volumes  (73  units  of  weight)  of  hydro- 
chloric acid  gas,  and  the  density  (i.e.  the  weight  of  unit 
volume)  of  this  will  be  equal  to  the  sum  of  the  com- 
bining (atomic)  weights  of  the  constituents  (i.e.  35.5  +  1 
=  36.5  units  of  weight).  And  this  will  also  be  true  for 
water  vapor  and  phosphine  when  we  use  the  factor  2 
for  oxygen  and  4  for  phosphorus. 

Representing,  then  the  combining  weight  of  an  ele- 
ment by  its  symbol,  and  designating  by  a  sub-numeral 
the  number  of  these  combining  weights  which  will  occupy, 
under  standard  conditions,  the  unit  of  volume,  we  shall 
obtain  the  formula  of  -the  element,  and  the  weight 
thus  represented  by  this  formula  will  occupy  the  same 
volume,  under  like  conditions,  as  the  formula  weight 
of  any  other  element  or  compound.  Applying  this  to 
the  three  cases  just  considered,  we  obtain 


where  although  the  weights  represented  by  the  terms 
H2,  C12,  O2,  P4,  HC1,  H2O,  and  PH3,  i.e.  the  formula 
weights,  are  all  different,  the  volumes  occupied  by  them 


10  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

are  identical,  provided  the  conditions  of  pressure  and 
temperature  are  the  same. 

But  these  are  the  formula  weights,  i.e.  are  what,  accord- 
ing to  hypothesis,  have  been  and  are  designated  as  molec- 
ular weights,  although  here  they  have  been  arrived  at 
wholly  without  the  aid  o)  hypothesis. 

Since  our  customary  unit  of  weight  is  the  gram,  and 
the  combining  weight  is  usually  based  upon  oxygen 
(  =  16),  we  can  define  a  formula  (molecular)  weight  in 
the  gaseous  state  as  that  weight  which  will  occupy  the 
volume  oj  2X16  grams  0}  oxygen  under  like  conditions. 
From  this  definition,  however,  it  is  at  once  obvious 
that  equal  volumes  o)  gas  under  like  conditions  must 
contain  the  same  number  0}  formula  weights.  But  this, 
in  its  practical  meaning,  is  identical  with  what  has  long 
been  known,  from  its  hypothetical  origin,  as  Avagadro's 
hypothesis,  which  now,  having  been  derived  from  experi- 
mental results,  without  the  aid  of  any  hypothesis,  becomes 
Avagadro's  law. 

As  has  been  mentioned,  the  factor  necessary  to  trans- 
form the  combining  (atomic)  weight  of  an  element  into 
the  formula  weight  according  to  this  definition  (i.e. 
the  number  of  combining  weights  which  will  occupy 
the  normal  volume  under  standard  conditions)  varies 
for  the  different  elements.  These  factors,  however,  are 
usually  small,  ranging  from  i,  for  most  of  the  gaseous 
metallic  elements,  to  8  for  gaseous  sulphur  under  cer- 
tain conditions.* 

We  have  arrived  thus  at  a  purely  experimental  hypoth- 
esis-free conception  of  the  atomic  and  molecular  weight 

*  For  a  list  of  these  factors,  and  a  similar  derivation  of  these  con- 
cepts, see  Ostwald,  Grundriss  der  allgemeinen  Chemie,  1899,  pp.  65-73. 


SOME  FUNDAMENTAL  PRINCIPLES.  H 

of  a  substance  in  the  gaseous  state.  It  is  not  to  be 
assumed,  though,  that  the  formula  (molecular)  weight 
necessarily  remains  the  same  in  the  other  states  of  aggre- 
gation, although  this  is  true  for  the  combining  weight, 
so  far  as  we  know.  Experiment  shows,  indeed,  that 
the  formula  weight  depends,  even  in  the  gaseous  state, 
according  to  this  definition,  upon  the  temperature  (sul- 
phur for  example) ;  while  in  the  state  of  solution  (accord- 
ing to  an  experimental  definition  derived  later)  it  often 
depends  upon  the  nature  of  the  solvent  (acetic  acid  in 
benzene  and  in  water) ;  and  in  the  liquid  state  upon 
the  presence  of  another  liquid  (alcohol  alone  and  with 
water.) 

Energy. — Energy  is  work  or  anything  which  can  be 
transformed  into  work  or  produced  from  work.  Although 
energy  may  appear  in  many  different  forms,  it  is  to  be 
remembered  that  all  these  forms  can  be  transformed, 
the  one  into  the  other.  The  principal  forms  under 
which  the  common  manifestations  of  energy  may  be 
grouped  are  as  follows :  Kinetic  energy,  i.e.  the  energy 
of  motion,  distance  or  potential  energy,  i.e.  the  energy 
of  position,  electrical  energy,  magnetic  energy,  heat, 
chemical  energy,  surface  energy,  volume  energy,  and 
radiant  energy.  But  since  these  forms  are  only  phases, 
as  we  may  say,  of  the  fundamental  concept  of  energy, 
and  each  can  be  transformed  into  the  other,  one  form,  viz. 
that  which  can  be  most  readily  defined,  has  been  chosen 
as  the  standard  of  reference.  In  other  words,  all  kinds  of 
energy  are  measured  and  expressed  in  terms  of  a  standard 
form,  into  which  they  could  all  be  transformed.  This 
standard  form  is  that  manifested  in  the  ordinary  mechan- 
ical relations. 

The  unit  of  work  is  the  erg,  which  is  the  work  done 


12  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

when  unit  force  is  overcome  through  unit  distance, 
the  unit  of  force  being  the  dyne,  i.e.  that  force  which, 
acting  for  i  second,  will  impart  to  i  gram  the  velocity 
of  i  centimeter  per  second.  Naturally,  then,  n  dynes 
will  impart  a  velocity  of  n  centimeters  per  second  to 
one  gram.  Since  at  Washington  a  body  falling  freely 
will  acquire  the  velocity  of  980.1  centimeters  per  second, 
the  force  of  gravitation  there  is  equivalent  to  980.1  dynes, 
and  the  work  of  raising  i  gram  through  i  centimeter 
is  980.1  ergs.  Or,  in  general,  the  force  of  gravitation 
is  equal  to  g  dynes,  and  the  work  of  raising  i  gram 
through  i  centimeter  is  g  ergs,  where  g  is  the  gravita- 
tional constant,  i.e.  the  velocity  per  second  acquired 
by  i  gram  in  falling  freely.  As  this  unit  is  exceedingly 
small,  and  results  expressed  in  it  are  cumbersome  to 
handle,  a  larger  unit,  i.e.  one  that  is  equal  to  ten  mil- 
lion ergs,  is  often  employed.  This  larger  unit  is  called 
the  joule  and  is  equal  to  io7  ergs,  while  a  still  larger 
one,  without  a  name  and  designated  by  J  (the  joule 
being  abbreviated  to  j)  is  equal  to  io10  ergs. 

Although  this  method  of  expressing  the  amount  of 
energy  involved,  independent  of  the  special  form  in 
which  it  is  at  the  time,  is  ideal,  it  is  not  in  general  use 
in  all  cases,  for  many  of  the  forms  of  energy  were  studied 
long  before  the  system  was  commonly  accepted.  For  this 
reason  we  still  find,  for  certain  forms  of  energy,  that  the 
older,  arbitary  units  are  still  in  vogue  (calorie,  coulomb, 
etc.).  One  unfortunate  consequence  of  this  older  no- 
menclature is  that  it  does  hot  keep  clearly  before  the 
mind  the  fact  that  the  forms  of  energy  are  but  phases 
of  the  general  concept  of  energy,  and  hence  are  mutu- 
ally capable  of  transformation.  To  accentuate  this  fact 
we  shall  always  give  the  value  of  these  arbitrary  units, 


SOME  FUNDAMENTAL   PRINCIPLES.  13 

Le.,  those  which  are  applicable  to  but  one  definite  form 
of  energy,  in  terms  of  the  general  standard. 

The  factors  of  energy.  —  Experience  has  shown  that 
the  total  amount  of  energy  of  any  kind  is  not  the  con- 
dition upon  which  its  transfer  from  one  place  to  another 
depends.  We  often  observe,  for  example,  that  energy 
is  transferred  from  a  system  containing  a  smaller  amount 
of  energy  to  one  containing  a  greater.  It  has  been  found 
possible,  however,  to  express  the  amount  of  each  form 
of  energy  as  a  product  of  two  factors,  one  of  which,  the 
so-called  intensity  factor,  conditions  the  transfer  of  that 
form  of  energy.  The  other  factor,  the  capacity  factor, 
although  having  nothing  to  do  with  the  transfer,  is  dis- 
tinguished by  the  fact  that  for  the  whole  system  it  remains 
constant  in  value,  i.e.,  the  transfer  of  the  energy  from 
one  system  to  another  leaves  the  capacity  factor  of  the 
whole  system  just  as  it  was  in  the  beginning,  viz.  equal  to 
the  sum  of  the  component  capacity  factors.  Whenever 
a  transfer  takes  place,  then,  there  has  previously  existed 
a  difference  in  intensity;  and  in  order  to  cause  a  transfer 
of  energy  to  take  place,  a  difference  in  intensity  is  essential. 

Examples  of  this  division  of  energy  into  factors  are 
very  numerous.  Thus  for  heat  energy  we  have  the  tem- 


perature T  as  the  intensity  factor,  entropy  being  the 


capacity  factor;  for  electrical  energy  (^X^o)  we  have 
the  electromotive  force  and  amount  of  electricity;  for 
volume  energy  (pXv)  the  pressure  and  volume;  while 
for  kinetic  energy  (%mc2)  we  may  have  either  velocity  and 
momentum,  or  the  square  of  the  velocity  and  the  mass. 
Two  bodies  at  the  same  temperature,  two  amounts  of 
electricity,  two  volumes  of  gas,  etc.,  will  always  remain 
in  equilibrium,  then,  i.e.  will  give  rise  to  no  transfer  of 


14  PHYSICAL  CHEMISTRY  FOR.  ELECTRICAL  ENGINEERS. 

energy  so  long  as  the  intensity  factors  (i.e.,  the  tempera- 
tures, the  pressures,  or  the  electromotive  forces)  are  the 
same. 

It  is  evident  since  the  intensity  factor  causes  the  trans- 
fer of  energy,  that  its  value  must  change  during  such 
a  transfer,  i.e.,  after  the  transfer  its  value  must  lie  be- 
tween the  two  original  intensity  values.  Thus  two  gases 
separated  by  a  movable  partition  will  cause  the  partition 
to  move  one  way  or  the  other,  according  to  which  body 
of  gas  has  the  greater  pressure,  and  the  final  pressure 
will  lie  between  the  two  initial  pressures.  The  volumes 
occupied  by  the  gases,  however,  will  have  nothing  to  do 
with  the  transfer  of  energy,  and  the  final  volume  of  the 
system  will  be  unaltered,  i.e.,  will  be  equal  to  the  sum 
of  the  two  initial  volumes.  An  especially  striking  exam- 
ple of  the  lack  of  influence  of  the  capacity  factor  in  the 
transfer  of  energy  is  given  by  a  system  made  up  of 
water  in  contact  with  air.  So  long  as  the  temperatures 
of  the  two  components  (water  and  air)  are  the  same, 
the  capacity  factors  may  be  altered  to  almost  any  extent 
by  decreasing  the  amount  of  air  and  increasing  that  of 
water;  but  alter  them  as  we  will,  no  heat  will  go  from 
the  water  to  the  air  or  vice  versa,  so  long  as  the  tempera- 
tures remain  the  same.  If  the  temperatures  be  different, 
however,  and  the  temperature  of  the  gas  be  higher  than 
that  of  the  water,  heat  will  immediately  go  from  the  air  to 
the  water,  notwithstanding  the  fact  that  the  amount  of  heat 
in  the  air  is  infinitesimal  as  compared  to  that  in  the  water. 

Expressing  these  facts  mathematically,  designating  the 
intensity  factor  by  the  letter  i,  and  the  capacity  factor  by 
c,  the  value  of  the  total  amount  of  energy,_E,  is  given  by 
the  equation 


SOME  FUNDAMENTAL  PRINCIPLES.  i$ 

Naturally,  when  both  c  and  i  are  allowed  to  vary,  the 
change  in  £  is  given  by  the  expression 

dE=cdi+idc, 

or,  retaining  one  constant  and  allowing  the  other  to  vary, 
by  either 

AE=cAi  (c  is  constant), 
or  AE=iAc  (i  is  constant), 

where  A  represents  a  finite  increase. 

When  there  are  two  forms  of  energy  active  in  a  system, 
and  a  change  in  one  produces  a  corresponding  change 
in  the  other,  we  have 


.e.,  c\    i  =  c^i. 

This  equation  is  exceedingly  valuable  hi  deriving  the 
relation  existing  between  the  two  kinds  of  energy  (heat 
and  electrical  energy,  for  example)  in  a  system,  and  we 
shall  have  occasion  to  make  use  of  it  in  our  later  work. 


CHAPTER  II. 

THE  GENERAL  PROPERTIES  OF  GASES. 

The  gas  laws. — In  spite  of  the  intangibility  which 
characterizes  the  gaseous  state,  and  the  consequent  dif- 
ficulty in  the  investigation  of  it,  our  knowledge  of  the 
laws  governing  the  behavior  of  gases  is  far  more  com- 
plete than  is  that  of  the  laws  regulating  the  behavior 
of  substances  in  the  other,  more  tangible,  states.  It  is 
not  to  be  assumed  from  this  statement,  however,  that 
we  know  why  gases  behave  as  they  do,  for  that  is  just 
what  we  do  not  know.  The  gas  laws  simply  state  how 
substances  in  the  gaseous  state  will  behave  under  those 
conditions  which  can  influence  them;  and  have,  and 
can  have,  nothing  to  do  with  the  question  as  to  the  cause 
of  this  behavior. 

Indeed,  the  difference  between  our  hypothesis-free 
standpoint  and  one  that  retains  hypotheses  is  very  well 
illustrated  by  the  difference  in  meaning  of  these  two 
words,  how  and  why.  To  say  how  a  thing  will  behave 
we  need  only  be  familiar  with  the  thing,  and  the  more 
familiar  we  are  with  it  the  more  accurate  will  be  our 
prediction.  In  other  words,  we  can  say  how  a  thing 
will  behave  under  any  condition  by  citing  facts  that 
have  been  observed,  or  an  experimental  law,  i.e.  a  gener- 
alization of  such  facts.  To  be  able  to  say  why  a  thing 
behaves  as  it  does,  on  the  other  hand,  it  is  necessary  to 

16 


THE  GENERAL  PROPERTIES  OF  GASES.  17 

go  beyond  the  facts  themselves,  and,  as  the  facts  form 
the  total  of  our  knowledge,  assume  that  a  certain  struc- 
ture, for  example,  is  responsible  for  the  behavior.  But, 
since  in  our  assumption  we  have  gone  beyond  the  facts, 
we  cannot  prove  it  to  be  either  correct  or  false  by  aid 
of  the  facts.  It  is  true,  of  course,  that  the  assumption 
may  satisfactorily  account  for  the  facts  as  we  know 
them  at  the  time,  but  we  have  no  reason  to  believe  that 
it  is  the  only  one  that  will  account  for  them,  or  even 
that  it  will  account  for  those  which  are  yet  to  be  dis- 
covered. In  other  words,  the  hypothesis  at  best  enables 
us  to  see  how  the  facts,  or  a  certain  number  of  them, 
might  be  explained ;  but  it  does  not  add  anything  of  value 
to  what  has  already  been  deduced  from  the  facts  them- 
selves, nor  does  it  lead  more  than  they  do  to  the  discov- 
ery of  new  facts.* 

This  repetition  of  our  standpoint  and  its  advantages 
is  quite  necessary  here,  for  no  portion  of  chemistry  is 
richer  in  hypothetical  assumptions  than  that  which 
includes  the  gaseous  state,  and  in  no  place  is  a  distinc- 
tion between  hypothesis  and  fact  more  essential.  In 
order  that  we  may  always  retain  our  standpoint,  and 
devote  ourselves  exclusively  to  facts,  therefore,  we  shall 
have  to  continually  inquire  as  to  how  things  occur,  avoid- 
ing any  assumption  as  to  why  they  should  occur  as  they  do. 

Although  the  gaseous  state  as  such  has  but  little  im- 
portance in  electrochemistry,  it  is  absolutely  essential 
that  the  electrochemist  obtain  a  clear  idea  of  its  laws, 
if  he  is  to  understand  the  laws  which  have  been  found 
to  govern  the  behavior  of  substances  in  solution.  And 
certainly  no  one  can  question  the  value  of  these  laws  to 

*  See  Ostwald's  Vorlesungen  iiber  Naturphilosophie,  I.e. 


18  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

the  electrochemist,  for,  with  but  few  exceptions,  all 
electrochemical  processes  take  place  in  solutions. 

Since  gases  are  characterized  by  an  absence  of  form, 
i.e.  occupy  entirely  any  space  in  which  they  are  present, 
their  condition  is  dependent  solely  upon  external  influences. 
Of  the  variations  which  a  gas  may  undergo,  as  the  result 
of  a  change  in  these  influences,  the  most  important  are 
those  of  volume,  pressure,  and  temperature.  Experiment 
shows,  however,  that  when  the  temperature  is  retained 
constant,  the  volume  of  the  gas  is  the  greater  the  smaller 
its  pressure,  and  vice  versa;  and,  so  long  as  the  volume 
(pressure)  remains  unchanged,  the  pressure  (volume)  is  the 
greater  the  higher  the  temperature,  and  vice  versa.  These 
laws  are  the  generalization  of  the  facts  observed  by 
several  generations  of  investigators,  and  an  exception  to 
them  is  yet  to  be  found. 

In  quantitative  work,  however,  such  purely  qualitative 
laws  are  of  little  value,  for  they  simply  indicate  the 
direction  of  the  variation  without  at  all  showing  its 
extent.  And  when  altered  to  show  the  extent,  they  lose 
the  absolute  generality  which  distinguishes  them,  and 
hold  true  with  accuracy  only  between  certain  limits. 

Quantitative  experiments  on  gases  have  resulted  in 
the  following  conclusions:  When  the  temperature  is 
retained  constant,  the  volume  is  inversely  proportional  to 
the  pressure  (Law  of  Boyle).  And  retaining  the  volume 
(pressure)  constant  the  increase  in  pressure  (volume)  per 
degree  centigrade  is  1/273  of  its  value  at  o°  centigrade 
(Law  of  Charles). 

Starting  with  unit  volume  at  o°,  and  constant  atmos- 
pheric pressure,  then,  and  decreasing  the  temperature  to 
—  273°  centigrade,  the  volume  of  the  gas  will  be  reduced 
to  zero  if  the  law  of  Charles  holds  at  such  a  temperature. 


THE  GENERAL  PROPERTIES  OF  GASES.  19 

In  other  words,  for  each  decrease  of  i°  the  volume  will 
be  reduced  1/2?z  of  its  original  value,  and  at  —273°  the 
loss  in  volume  will  be  273/273,  i.e.,  i.  If  we  consider 
—  273°  centigrade  as  the  zero  of  a  new  scale  (the  abso- 
lute zero),  and  employ  centigrade  degrees,  calling  the 
temperatures  absolute  temperatures,  we  can  say — at 
o°  absolute  (i.e.,  —273°  centigrade)  the  volume  of  a  gas 
(as  a  gas)  is  zero,  and  its  volume  will  increase  per 
degree  by  1/273  of  the  value  it  would  have  at  273° 
absolute  (i.e.,  o°  centigrade),  and  that  increase  will 
always  be  the  same,  independent  of  the  actual  tempera- 
ture. 

Expressing  these  relations  mathematically  we  have 

•v  a  —  (T  is  constant), 

and  v<xT  (pis  constant); 

T 
I.e.  1>oc— , 


or  v-j, 

and  pv=kT. 

But  at  273°  absolute  (o°  centigrade) 

=k2  73, 


and  by  combination,  eliminating  the  common  constant  k, 
we  obtain 


273 


20  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

which  gives  the  value  of  the  product  pv  at  any  tempera 
ture  (absolute)  in  terms  of  p  and  v  at  273°  absolute. 


The  term  -    -  is  a  constant,  however,  which  depend 

273 

for  its  value  only  upon  the  weight  of  gas  occupying  th 
volume  v0,  and  the  units  of  volume  and  pressure  chosen 

Considering  v0  as  the  volume  of  i  gram  of  gas,  we  hav 


where,  although  the  value  of  r,  the  specific  gas  constani 
depends  upon  the  units,  it  is  constant  for  any  one  gas. 

Regarding  VQ  as  the  volume  occupied  by  i  mole  of  gas 
on  the  other  hand,  and  designating  it  by  F0,  we  fm< 

-  RT, 


273 

where  R,  the  molecular  gas  constant,  is  the  same  for  al 
gases,  for,  by  definition,  page  10,  i  mole  of  gas  alway 
occupies  the  normal  volume  under  standard  conditions. 

The  difference  in  meaning  of  these  two  constants, 
and  R,  accentuates  the  difference  in  standpoint  of  physic 
and  physical  chemistry.  While  the  results  of  physic 
are  always  given  for  a  specific  quantity  of  substance 
without  regard  to  the  similarities  which  chemistry  ha 
discovered  to  exist  between  the  formula  weights  of  al 
substances,  physical  chemistry  employs  physical  method: 
from  the  chemical  point  of  view,  i.e.  strives  to  disco  ve: 
general  laws  by  applying  both  physics  and  chemistry  tc 
the  facts  observed. 

Since  a  mole  of  any  substance  in  the   gaseous   stat< 


THE  GENERAL  PROPERTIES  OF  GASES.  21 

occupies  the  same  volume  as  32  grams  of  oxygen,  i.e. 
32X699.8=22393.6  cubic  centimeters,  or  nearly  22.4* 
liters,  at  o°  and  76  centimeters  pressure,  we  can  find  the 

value  of  R= for  the  various  units  in  which  pressure 

273 

and  volume  may  be  expressed.  As  the  specific  gravity 
of  mercury  is  13.6,  the  pressure  of  76  centimeters  is 
1033.6  grams  per  square  centimeter  (i.e.  13.6X76),  and 
we  have 


o     1033^6X22400 


273  273 

=  84800  (F  in  c.c.,  p  in  grams  per  sq.  cm.) 

=  848ooX 980.1  =  8.3  X  io~7(F  in  c.c.,  p  in  dynes) 

I  X22.4 


273 


=  0.0821  (F  in  liters,  p  in  atmospheres). 


Another  law  which  is  of  great  value  in  considering  the 
gaseous  state  is  that  of  Dalton,  according  to  which 
each  component  of  a  mixture  of  gases  exerts  the  same 
pressure  in  the  system  as  it  would  exert  were  it  alone  present 
in  the  volume  of  the  mixture.  In  other  words,  the  pres- 
sure of  a  system  composed  of  several  gases  is  an  addi- 
tive property. 

Summarizing  the  gas  laws  in  their  mathematical 
forms,  then,  we  have 


*  As  the  factor  to  transform  grams  to  ounces  (av.)  is  the  same  as 
that  for  the  transformation  of  liters  into  cubic  feet,  i  mole  of  gas  in 
ounces  (av.)  occupies  22.4  cubic  feet  at  o°  and  76  cms.  of  mercury 
pressure. 


22  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  EMG1HEERS. 
Vi:v2:  :p2:pi,  or  p  iVi  =  p2v2  =  const.  (T  is  constant), 
pi  :  p2:  :Ti:T2  (v  is  constant), 
vi  :  v2  :  :  T\  :  T2    (p  is  constant)  , 
pV=RT 

(where  R  is  constant  for  all  gases  and  depends  in  value 
upon   the  units   of  pressure   and  volume   chosen),  and 


where  P  is  the  total  pressure  of  a  mixture  of  gases  and  the 
terms  p  are  the  partial  pressures  of  the  components  of 
the  mixture. 

As  already  mentioned,  to  secure  quantitative  results  it 
is  necessary  to  restrict  the  limits  of  the  general  laws 
holding  for  gases.  Both  these  laws,  viz.  that  of  Boyle 
and  that  of  Charles,  hold  rigidly,  then,  only  between 
certain  definite  limits.  In  fact,  stated  as  above,  they 
hold  rigidly  only  for  ideal  gases.  It  has  been  found, 
however,  that  the  further  a  gas  is  removed  from  its  lique- 
faction point,  the  more  nearly  ideal  it  is;  i.e.,  the  more 
accurately  is  its  behavior  represented  by  the  laws.  In- 
deed, the  law  of  Boyle,  as  applied  to  the  so-called  perma- 
nent gases,  gives  very  satisfactory  results,  except  when  the 
pressure  is  very  high;  as  does  the  law  of  Charles,  so  long 
as  the  temperatures  are  not  excessive.* 

*  For  hydrogen  the  product  pV  (or  pv)  at  constant  temperature 
increases  steadily  and  regularly  with  an  increase  of  pressure,  and  is 
expressed  very  accurately  by  the  altered  form  of  the  equation  p(V—b) 
=  const.,  where  b  is  a  constant.  All  other  gases,  on  the  other 
hand,  starting  with  atmospheric  pressure  and  constant  temperature, 
give  a  value  of  pv,  which  first  decreases  with  increased  pressure,  then 
passes  through  a  minimum,  and  finally  increases  steadily.  The  rela- 
tions, in  such  a  case,  can  be  followed  by  aid  of  Van  der  Waal's  equation, 


THE  GENERAL  PROPERTIES  OF  GASES.  2 3 

Dissociation. — By  aid  of  the  relation  pV=RT  the 
so-called  equation  oj  state  jor  gases,  it  is  possible  to  obtain 
a  new  form  of  definition  for  the  formula  (molecular) 
weight  in  the  gaseous  state.  Instead  of  defining  it  as 
the  weight  which  under  like  conditions  will  occupy  the 
volume  of  32  grams  of  oxygen,  we  may  say  the  formula 

pV 
weight  of  a  gas  is  the  weight  which  will  give  V,  in  -=-  =  R, 

such  a  value  that  R  is  approximately  equal  to  that  cal- 
culated for  oxygen. 

Neither  of  these  methods,  however,  is  adapted  to  labora- 
tory requirements.  In  practical  work  it  is  much  simpler 
to  take  advantage  of  the  fact  (p.  10)  that  equal  volumes 
of  gas  contain  the  same  number  of  formula  weight  (moles), 
and  to  determine  the  density  of  the  gas  in  terms  of  oxygen. 
Since  the  formula  weight  of  oxygen  is  32,  the  formula 
weight  of  the  gas  can  then  be  obtained  directly  by 
multiplying  the  density  ratio  by  32;  i.e., 

weight  of  i  c.c.  of  gas 

M  =  32  X :— TI — £ 1 > 

weight  of  i  c.c.  of  oxygen 

the  conditions  of  temperature  and  pressure  being  the 
same. 

For  certain  substances,  however,  the  formula  weight 
so  determined  is  found  to  vary  with  the  temperature 
and  pressure,  and  in  some  cases  to  be  equal  to  but 
one-half  the  sum  of  the  combining  weights  of  the  con- 

i.e.,  ( p  +  y-2J  ( V—  b}  =  const.,  where  a  and  b  are  constant  values  depend- 
ing only  upon  the  nature  of  the  gas.  In  the  case  of  ethylene,  for 
example,  the  results  by  this  formula  agree  very  accurately  with  the 
experimentally  determined  ones  up  to  a  pressure  of  400  atmospheres. 
("Elements,"  pp.  34-38.) 


24  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

stituents.  Such  a  result  at  first  glance  would  natu- 
rally lead  one  to  object  to  the  process  of  reasoning  em- 
ployed on  page  9,  and,,  indeed,  without  further  knowl- 
edge of  the  substances  behaving  in  this  way,  it  would 
be  impossible  to  defend  its  use.  It  is  obvious,  then, 
that  either  our  definition  of  a  formula  weight  in  the 
gaseous  state  (p.  10)  is  incorrect  in  some  cases,  or  that 
some  process — common  to  these — is  responsible  for  the 
abnormal  results  observed. 

Further  investigation  of  these  substances,  however, 
shows  conclusively  that  our  definition  is  correct  in  all 
cases,  and  that  the  abnormal  results  are  due  to  a  more 
or  less  complete  decomposition  of  the  substance,  a  dis- 
sociation, as  it  is  called. 

Of  the  substances  for  which  such  abnormal  results 
are  observed  we  may  mention  ammonium  chloride 
(NH4C1),  phosphorus  pentachloride  (PC15),  and  nitrogen 
tetroxide  (N2OJ  as  typical  examples,  although  many 
others  might  be  cited.  The  formula  weights  (by  defi- 
nition) of  these  substances  have  each  been  observed  to 
vary  from  the  normal  value  represented  by  the  formula 
to  a  minimum  value  which  is  one-half  of  this,  the  amount 
of  the  variation  depending  upon  the  temperature  and 
pressure.  In  general,  the  higher  the  temperature  under 
constant  pressure  the  lower  the  value,  while  at  constant 
pressure  the  formula  weight  is  the  lower  (one-half  of  the 
formula  weight  being  the  limit)  the  lower  the  pressure. 

The  very  fact  that  a  general  rule  of  this  sort  exists 
for  substances  behaving  abnormally  is  evidence  that 
it  is  not  the  definition  of  the  formula  weight  which  is 
at  fault,  but  that  some  process  occurring  similarly  in 
all  these  cases  is  responsible  for  a  change  in  the  formula 
weight  itself.  Indeed,  later  work,  by  which  the  actual 


THE  GENERAL  PROPERTIES  OF  GASES.  2$ 

presence  of  the  products  of  the  decomposition  was  shown, 
proved  that  the  definition  of  a  formula  weight  even  in 
these  cases  is  correct  and  that  it  was  only  our  ignorance 
of  the  effects  of  temperature  and  pressure  upon  these 
substances  which  led  us  to  infer  otherwise.  Instead 
of  destroying  the  usefulness  of  one  of  our  fundamental 
principles,  then,  these  "  abnormal  "  results  have  simply 
introduced  to  our  attention  a  very  important  and  com- 
mon process,  viz.,  that  of  dissociation. 

In  the  three  cases  mentioned  above  the  dissociation 
has  been  found  to  take  place  according  to  the  following 
schemes,  the  symbol  +±  being  used  to  show  that  the 
reaction  is  reversible,  i.e.  that  it  goes  in  one  direction 
or  the  other,  depending  upon  the  conditions: 

NH4C1^NH3+HC1, 


N204^N02+N02. 

The  presence  of  the  NH3,  HC1,  Cl,  and  NO2  in  these 
cases  can  be  proven  without  difficulty.  The  gas  which 
is  evolved  from  ammonium  chloride  by  heat,  for  example, 
can  be  shown  to  contain  NH3  and  HC1  by  allowing  it 
to  diffuse  through  a  porous  diaphragm.  Here  the  NH3 
being  the  lighter  diffuses  more  rapidly  than  the  HC1, 
and  an  excess  of  NH3  is  found  on  one  side  of  the  par- 
tition, while  an  excess  of  HC1  remains  on  the  other. 
The  presence  of  chlorine  in  PC15,  and  NO2  in  N2O4,  is 
even  easier  to  show,  for  the  dissociation  can  be  followed  by 
the  eye,  the  chlorine  imparting  a  green,  the  NO2  a  brown- 
ish red,  color. 

These  methods,  together  with  the  many  others,  although 


26  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

proving  conclusively  that  a  decomposition  (dissociation) 
does  take  place,  are  of  little  value  in  determining  the 
extent  to  which  it  takes  place,  for  at  best  they  are  but 
qualitative.  For  this  purpose  the  simplest  and  most 
accurate  method  is  based  upon  the  density.  It  will  be 
observed  that  in  each  of  the  cases  above  complete  dis- 
sociation would  transform  one  formula  weight  (mole) 
into  two.  The  volume  occupied  by  the  two  formula 
weights,  however,  will  be  double  that  which  would 
contain,  under  like  conditions,  the  formula  weight  of 
the  undissociated  substance,  and  the  density  (i.e.  the 
weight  of  unit  volume)  after  complete  dissociation  will 
be  one-half  what  it  would  be  without  dissociation.  When 
the  dissociation  is  not  complete  we  can  also  follow  it 
and  determine  its  extent  from  the  two  densities,  i.e. 
with  and  without  dissociation,  although  the  relation  is 
not  so  simple. 

Assume  in  a  case  such  as  the  above  that  we  start  with 
i  mole  of  the  undissociated  state,  and  that  this  dis- 
sociates to  the  extent  of  oc%.  From  the  one  mole  before 
dissociation,  then,  we  obtain  i  +a  moles  after  dissocia- 
tion, for  we  have  i  —  a  still  undissociated,  and  a  each 
of  the  two  products,  i.e.  i—  a  +  2a  =  i+a  moles  in 
total.  Under  like  conditions,  the  volumes  occupied 
by  the  substance  before  and  after  dissociation,  then, 
will  be  related  as  III+CK;  and  the  densities  before  and 
after  will  be  related  as  i  +a :  i,  for  the  greater  the  volume 
produced  by  the  dissociation  of  the  original  mole  the 
smaller  will  be  the  density  of  the  system.  In  general, 
consequently,  for  all  substances  forming  2  moles  from 
i  by  dissociation  we  have  the  relation 


THE  GENERAL  PROPERTIES  OF  GASES.  27 

where  dd  is  the  observed  density  after  dissociation,  and 
du  is  the  density  without  dissociation,  both  being  meas- 
ured under  the  same  conditions  of  pressure  and  tem- 
perature. The  term  du  can,  naturally,  be  obtained  from 
the  formula  weight  of  the  undissociated  substance,  i.e. 
is  approximately  one-half  the  formula  weight,  when  based 
upon  hydrogen,  or  one  thirty-second  when  based  upon 
oxygen. 

When  one  mole  of  substance  produces  three  moles 
by  complete  dissociation,  for  example  the  case  of  ammo- 
nium carbamate,  i.e. 

NH2CO2NH4^C02  +  2NH3, 

and  a  is  the  degree  of  dissociation,  we  have  (i  —  a)  -f- 
3a:i::du:dd.  And,  in  general,  where  i  mole  falls  into 
n  moles  by  dissociation, 

I-OL  +  na:i:  :d:d 


ud, 


or  a 


(n-i)dd9 


from  which  a  may  be  calculated  without  difficulty. 

It  is  not  to  be  assumed  because  we  have  restricted 
ourselves  to  these  few  typical  examples  that  the  process 
of  dissociation  is  simply  a  scientific  curiosity,  for  it  is 
not  only  exceedingly  important  in  itself,  but  is  abso- 
lutely essential  to  things  we  shall  have  to  consider  later. 
Without  knowing  it,  in  fact,  the  reader  has  probably 
made  use  of  the  process  of  gaseous  dissociation  in  quali- 
tative analysis,  for  the  Marsh  test  for  arsenic  depends 
solely  upon  it.  The  gaseous  arsine,  which  is  dissociated 
by  its  passage  through  the  red-hot  tube,  re-unites  in  the 
colder  portions,  but,  since  the  hydrogen  diffuses  more 


28  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

rapidly  than  the  arsenic,  an  excess  of  the  latter  is  left 
behind,  and  crystallizes  upon  the  walls. 

In  addition  to  the  influence  of  pressure  and  temperature 
upon  .the  degree  of  dissociation  of  a  gaseous  substance, 
it  has  been  found  that  the  introduction  of  one  of  the  prod- 
ucts o)  the  dissociation  into  the  system,  either  before  or 
after  the  process  takes  place,  always  decreases  the  degree 
of  dissociation  of  the  substance.  Although  the  calculation 
of  the  effect  of  these  three  influences  upon  the  dissocia- 
tion can  only  be  considered  later,  it  may  be  mentioned 
here  that  a  formula  can  be  derived  by  which  both  the 
effect  of  pressure  and  that  of  the  addition  of  a  definite 
amount  of  a  product  may  be  calculated  at  constant 
temperature,  and  that  another  formula  enables  us  to 
follow  the  influence  of  temperature. 

The  following  tables  will  enable  the  reader  to  gather 
some  idea  of  the  effect  of  pressure  and  temperature  upon 
the  dissociation,  and  also  make  clear  the  calculation  of 
a  in  the  various  cases. 

DISSOCIATION  OF  NITROGEN  TETROXIDE,  N2O4. 
(Density  of  N2O4=  3.18;  of  NO2+NO2=  1.59;  air=i.)* 


Temp.                        Sp.  Gr.  of  Gas. 

Percentage  Dissociation. 

26°.  7                          2.65 

19.96 

35°-4                             2.53 

25-65 

39°.  8                            2.46 

29.23 

49°.  6                           2.27 

40.04 

60°.  2                                         2.08 

52.84 

70°.  o 

.92 

65.57 

80°.  6 

.80 

76.61 

90°.  o 

.72 

84.83 

100°.  I 

.68 

89.23 

in0.  3 

•65 

92.67 

121°.  5 

.62 

96.23 

135°-° 

.60 

98.69 

154°.  o                              .58 

I  OO  .  OO 

*  The    densities    of    hydrogen,    oxygen,    and    air    are   related    as 
: :  15.88: 14.45,  from  which  the  densities  in  other  units  may  be  calculated. 


THE  GENERAL  PROPERTIES  OF  GASES.  29 

DISSOCIATION  OF  PC16. 

(Density  PC15=  7.2;  PC13+  Cl,=  3.6;  air=i.) 
Temy.                              Density.          Percentage  Dissociation. 

182°                                 5.08  41-7 

190°                                4.99  44-3 

200°                                4.85  48-5 

230°                                4.30  67.4 

250°                             4 . oo  80 . o 

274°                             3-84  87.5 

288°                             3.67  96.2 

3°°°                          3-6S  97-3 


DISSOCIATION  OF  N2O4. 
(Equal  Temperatures,  Varying  Pressures.) 

Temp.  Pressure.  Density  (air  =  i). 

i8°.o  279.0  mm.  2.71  17.3 

i8°.s  136.0     "  2.45  29.8 

20°. o  301.0     "  2.70  17.8 

20°. 8  153.5     "  2-46  29.3 

When  i  mole  of  gas  is  formed  from  a  solid  or  liquid, 
at  the  constant  pressure  p,  the  work  done  is  equal  to 
p  times  the  increase  in  volume.  Since  the  volume  of 
the  solid  or  liquid  is  negligibly  small,  as  compared  to 
that  of  a  gas,  however,  we  may  regard  the  total  volume 
occupied  by  the  gas  as  equivalent  to  the  increase  of  volume. 
The  work  done,  then,  will  be  equal  to  pV.  But  the 
product  pV  at  constant  temperature  is  constant,  inde- 
pendent of  the  countless  values  of  p  and  V  from  which  it 

might  be  made  up  ( for  />oc— j,  and  is  equal  in  value 

jor  i  mole  to  RT,  in  which  R  is  a  constant,  energy  quantity. 
In  calculating  amounts  of  work  of  this  kind,  then,  we 
shall  always  employ  the  right  side  of  the  equation  pV=RT, 
for  it  shows  the  relations  much  more  definitely  and 


30  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

clearly  than  the  other.  Thus  to  form  i  mole  of  gas  at 
any  temperature  against  any  pressure  always  requires 
the  work  RT,  which  will  be  given  in  the  units  employed 
for  R,  since  T  is  a  pure  number. 

If  instead  of  forming  one  mole,  as  we  suppose,  the 
gas  dissociates  into  others,  the  work  involved  will  still 
be  RT  units  for  each  mole,  but  iRT  in  total,  where  i 
is  equal  to  the  number  of  moles  formed  from  one  original 
mole.  This  value  i  may  also  be  expressed  in  terms  of 
the  degree  of  dissociation,  for,  if  this  is  a,  and  n  moles 
are  formed  from  one,  i,  the  total  number  of  moles,  will 
be  (i  —a)  +na,  and  the  work  of  formation  from  a  liquid 
or  solid  will  be  [(i  —a)  +na]RT units. 

Naturally,  we  can  also  define  a  formula  weight,  in 
the  gaseous  state,  in  such  terms.  The  formula  weight 
is  then  that  weight  of  gas  which  in  forming  from  a  liquid 
or  a  solid  (i.e.,  from  a  volume  which  is  negligible)  will 
perform  approximately  the  same  amount  of  work  as  32 
grams  of  oxygen  would,  i.e.,  RT  units,  at  the  tempera- 
ture T,  against  any  pressure.  We  could  also  determine 
dissociation  in  this  way,  i.e.,  by  determining  the  work 
done  and  finding  the  value  of  i  as  a  difference,  from  which 
a,  when  n  is  known,'  can  be  found;  but  the  method 
based  upon  the  density  is  far  simpler  and  decidedly 
more  practical. 

Partial  pressures  and  concentrations. — For  the  descrip- 
tion of  a  gaseous  system  composed  either  of  a  single 
gas  or  a  mixture  of  gases  it  is  essential  that  we  have 
a  convenient  method  of  representing  the  amount  of  a 
gas  present.  This  is  not  only  necessary  for  the  descrip- 
tion of  such  a  system,  and  for  use  in  the  formula  (mentioned 
above)  which  shows  the  dependence  of  the  degree  of 
dissociation  upon  the  pressure,  and  upon  the  amount  of 


THE  GENERAL  PROPERTIES  OF  GASES.  31 

one  of  the  dissociation-products  that  has  been  added  to 
the  system,  but  will  also  serve  to  simplify  some  of  our 
later  work. 

Thus  far  we  have  used  either  the  density  of  the  gas 
or  the  volume  (F)  which  contains  i  mole  for  this  pur- 
pose, but  these  are  not  the  only  forms  of  expression,  nor 
are  they  even  the  most  convenient.  We  shall  therefore 
consider  briefly  the  other,  better  mehods  for  the  definition 
of  the  amount  of  gas  present  in  a  system. 

Since  under  definite  conditions  i  mole  of  gas  (by  defi- 
nition) occupies  a  definite  volume,  and  i  liter  of  this  vol- 
ume will  contain  a  definite  fraction  of  a  mole,  under 
those  conditions,  it  is  evident  that  any  change  in  the  system 
due  to  an  alteration  of  the  conditions  can  be  accurately 
described  by  a  statement  of  the  change  in  concentration, 
i.e.,  the  change  in  the  number  of  moles  per  liter.  But, 
at  constant  temperature,  the  concentration  is  propor- 
tional to  the  pressure,  i.e.,  the  greater  the  concentration 
the  greater  the  pressure,  and  vice  versa,  so  that  it  is 
obvious  that  a  change  in  any  gaseous  system  can  also 
be  accurately  described  by  a  statement  of  the  change  in 
pressure  (or  partial  pressures  in  case  the  system  is  com- 
posed of  several  gases). 

As  these  two  terms,  concentration  and  partial  pressure, 
are  to  be  used  constantly  in  our  later  work,  it  will  be 
necessary  here  to  find  the  exact  quantitative  relation 
which  exists  between  them.  From  the  definition  of 
the  formula  weight,  however,  remembering  the  laws  of 
Boyle  and  Charles  (p.  18),  this  relation  follows  directly, 
provided  the  laws  continue  to  hold.  For  at  o°  C.  the 
concentration  i,  i.e..  j  mole  per  liter,  is  equivalent  to  a 


pressure  of  22.4  atmospheres  [P^-y],  and  at  any  other 


32  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

absolute  temperature  T,  the  unit  of  concentration  gives 

T 

the  pressure  22.4 —  atmospheres. 

In  order  that  the  reader  may  become  familiar  with  the 
use  of  these  terms,  we  shall  now  consider  a  specific  case 
of  the  system  formed  by  dissociation.  At  190°  C.,  for 
example,  we  find  that  PC15  is  44.3%  dissociated  (p.  29), 
i.e.,  the  reaction 


goes  toward  the  right  until  44.3%  of  the  PC15  originally 
present  is  decomposed  and  equilibrium  is  established. 
We  have  then  a  system  composed  of  the  three  gases, 
PC15,  PC13,  and  chlorine.  Since  the  data  in  the  table 
(p.  29)  are  given  for  atmospheric  pressure,  the  total 
pressure  after  dissociation  remains  atmospheric  and  the 
volume  increases.  Starting  with  i  mole  of  PCls,  assum- 
ing no  dissociation  to  take  place,  the  volume  would  be 

22.4  -    -   liters.     But   by   the   dissociation   we  lose 

0.443  m°le  °f  PC15,  and  gain  °-443  of  a  mole  of  each 
of  the  products,  PCla  and  chlorine,  hence  the  total  volume 

after  dissociation  is  1.443X22.4—  -  —  liters,   for  the 

mole  of  PC15  has  been  transformed  into  (i  -0.443)  + 
2X0.443  =  1.443  moles,  the  temperature  and  pressure 
remaining  unchanged.  The  partial  pressures  in  atmos- 


pheres, then,  will  be  -  ~°'443  for  PC15,  and  each 

x-443  1-443 

for  PCla  and  chlorine,  i.e.,  the  number  of  moles  of  each, 
divided  by  the  total  number  of  moles  in   the   system, 


THE  GENERAL  PROPERTIES  OF  GASES.  33 

and  multiplied  by  the  total  pressure.    And  the  concen- 
trations, i.e.,  number  of  moles  per  liter,  will  be 

i  -0.443 


273  +  190      Pa»' 
1.443X22.4-^^ 


Q-443 

chlorine> 


1.443X22.4— 


i.e.  the  number  of  moles  divided  by  the  total  volume 
in  liters. 

In  case  the  volume  remained  constant,  i.e.,  that  the 
pressure  increased  by  the  dissociation  (assuming  the 
increase  of  pressure  to  have  no  effect  upon  the  degree 
of  dissociation)  the  final  pressure  would  be  1.443  atmos- 
pheres. The  partial  pressures  hi  atmospheres,  then, 
would  be  i  —0.443  f°r  PCls,  and  0.443  eacn  ^o 
and  chlorine.  And  the  concentrations  would  be 

i  -Q.443 

*"* 


and 

0-443 


273  +  190 

22.4 

273 


~~  CPC13  —  ^chlorine- 


CHAPTER  III. 

HEAT  AND  ITS  TRANSFORMATION  INTO  OTHER  FORMS 
OF  ENERGY. 

The  first  law  of  thermodynamics. — As  has  already, 
been  mentioned,  energy  may  be  manifested  in  various 
forms.  An  amount  of  energy,  then,  may  be  expressed 
in  either  the  so-called  absolute  units  (erg,  dyne,  etc.), 
or  in  any  of  the  units  employed  exclusively  for  one  specific 
form  of  energy  (coulomb,  calorie,  etc).  This  conception 
of  energy  is  the  one  which  has  been  commonly  accepted 
since  the  time  of  J.  R.  Mayer  (1841),  who  was  the  first 
to  determine  the  factor  necessary  to  transform  energy 
expressed  as  heat  into  mechanical  units,  i.e.,  the  so-called 
mechanical  equivalent  of  heat.  It  is  to  be  remembered 
here,  however,  that  the  word  equivalent  is  used  only 
in  the  sense  that  if  heat  is  transformed  into  mechanical 
work  we  shall  always  obtain  a  definite  number  of  mechan- 
ical units  from  each  unit  of  heat  (calorie)  transformed, 
and  does  not  at  all  imply  that  heat  is  always  transformed 
into  work  under  all  conditions,  or  that  all  the  heat  present 
will  be  transformed  into  work.  The  principle  governing 
the  transfer  of  heat,  and  the  relation  existing  between 
the  heat  transferred  and  that  transformed  into  work,  is 
to  be  considered  below;  here  we  shall  only  discuss  the 
relation  of .  the  amount  of  heat  actually  transformed 
into  mechanical  work  to  the  amount  of  work  which 
results  from  the  transformation. 

34 


HEAT  AND  ITS  TRANSFORMATION.  35 

Of  the  various  methods  for  determining  the  mechanical 
equivalent  of  heat  (Joule's,  Rumford's,  etc.),  the  most 
interesting  in  principle  for  our  purposes  is  that  of  J.  R. 
Mayer,  which  was  the  forerunner  of  all  the  others.  This 
depends  upon  the  difference  observed  when  the  specific 
heat  of  a  gas  is  determined  for  variable  or  for  constant 
volume.  Since  by  definition  the  specific  heat  of  a  sub- 
stance is  the  heat  necessary  to  raise  i  gram  of  it  i°  C. 
(from  some  one  standard  temperature),  it  is  obvious 
that  for  gases  two  values  will  be  found,  one  when  the 
gas  is  allowed  to  expand  under  constant  pressure,  the 
other  when  the  volume  is  retained  constant.  Natu- 
rally, d  priori,  there  is  no  means  of  deciding  whether 
these  two  values  will  be  experimentally  identical  or 
not,  unless  we  understand  the  difference  in  the  process 
in  the  two  cases.  Experiment,  however,  shows  the 
values  to  be  different  when  work  is  done  by  the  expan- 
sion, the  one  for  constant  pressure  (i.e.  where  the  volume 
increases)  being  the  larger;  while  no  difference  from 
that  at  constant  volume  is  observed  when  the  expansion 
takes  place  without  involving  mechanical  work,  e.g.  into 
an  exhausted  space.  Mayer  was  the  first  to  recognize 
that  the  only  difference  possible  in  these  two  experimental 
values  is  the  amount  of  heat  which  is  used  in  over- 
coming the  resistance  offered  to  expansion.  From  actu- 
ally observed  values,  then,  it  was  possible  for  him  to  cal- 
culate the  exact  value  of  a  calorie  in  mechanical  units. 
By  experiments  with  air,  for  example,  it  is  found 
that 

cp  —  cw= 0.0692  cal., 

i.e.,  the  amount  of  work  necessary  to  expand  i  gram 
of  air  by  the  1/273  d  part  of  its  volume  at  o°  C.,  against 


36  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

the  pressure  of  the  atmosphere,  is  equivalent  to  0.0692 
cal.  Since  i  gram  of  air  occupies  773.3  c.c.,  it  would 
occupy  773.3  cm.  of  a  tube  having  a  cross-section  of 
i  square  centimeter.  An  increase  in  temperature  of 


i°,  then,  would  involve  an  increase  of  2.83  ( i.e.  ^-^  \  cm<j 


and  the  weight  of  the  atmosphere,  13.6X76  =  1033.6 
grams  per  square  centimeter,  must  be  raised  through 
this  distance  of  2.83  cm.  But  the  work  necessary  to 
do  this  is  2.83X1033.6  =  2928  gram-centimeters,  and 
hence 

0.0692  cal.  =  2928  gram-centimeters, 

i.e.,     i  cal.  =42,300  gram-centimeters, 

which  is  the  mechanical  equivalent  of  heat.* 

It  has  been  shown  since  Mayer's  time  that  in  general 
any  form  of  energy  can  be  transformed  into  any  other 
form,  and  that  in  every  case  we  obtain  from  each  unit 
(which  has  been  transformed)  of  the  original  form  of 
energy  a  definite,  constant  number  of  units  of  the  final 
form  resulting  from  the  transformation.  Or,  expressing 
it  in  another  way,  if  we  start  with  an  amount  of  energy 
in  the  form  A  and  transform  it  completely  into  the  form  B, 
and  this  completely  into  the  form  D,  which  is  then  trans- 
formed completely  into  the  form  A  again,  the  original 
and  final  amounts  of  A  will  be  identical.  And  when 

*  Since  the  experimental  error  in  determining  the  specific  heat  of  a 
gas,  particularly  for  constant  volume,  is  not  inconsiderable,  this  result 
is  less  accurate  than  later  ones,  determined  in  other  ways,  although 
all  methods  give  sensibly  the  same  result.  The  commonly  accepted 
value  of  the  mechanical  equivalent  of  heat  at  present  is  42,600  gram, 
centimeters,  so  that  we  shall  use  it,  in  preference  to  the  above  value, 
in  all  our  later  calculations. 


HEAT  AND  ITS   TRANSFORMATION.  37 

in  any  case  the  transformation  is  not  complete,  another 
kind  of  energy  of  the  form  Zi,  Z2,  etc.,  being  produced 
to  a  slight  extent,  the  final  amount  of  A  will  be  smaller 
than  the  original  one  by  the  sum  of  the  amounts  of  Zi, 
Za,  etc.,  expressed  in  the  same  units  as  A.  Or  the 
final  value  of  A  plus  the  sum  of  the  amounts  of  Zi,  Z2, 
etc.,  after  they  are  completely  re-transformed  into  A,  will 
be  identical  with  the  initial  value. 

These  facts  concerning  the  mutual  transformation 
of  the  various  kinds  of  energy  are  usually  summed  up 
in  the  form  of  a  law  which  is  known  as  the  first  law 
o)  thermodynamics  or  energetics.  This  law  may  be  ex- 
pressed most  conveniently  in  one  of  the  two  following 
ways  :  The  energy  of  any  isolated  system  is  constant;  or,  a 
perpetual  motion  oj  the  first  kind  is  impossible,  i.e.  energy 
can  neither  be  created  nor  destroyed. 

If  a  system  is  heated,  then,  i.e.  absorbs  energy  from 
its  environment,  all  the  energy  which  it  has  gained  must 
appear  either  in  the  form  of  the  energy  absorbed  or  in 
some  other  form,  and  the  total  amount  of  energy  in  the 
system  will  be  equal  to  that  which  it  contained  originally 
plus  that  added,  both  being  expressed  in  the  same  terms. 
A  gaseous  system,  for  example,  when  heated,  will  increase 
in  temperature,  and  may  also  expand  against  a  pres- 
sure, i.e.  do  external  mechanical  work.  Expressed 
mathematically,  then,  the  first  law  of  energetics  as  applied 
to  the  system  will  lead  to  the  relation 


where  dE  is  the  energy  absorbed,  dU  is  the  increase  of 
internal  energy,  and  W  is  the  external  mechanical  work 
involved  in  the  expansion,  all  being  expressed  in  the 


38  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

same  kind  of  units.  The  term  W,  for  a  gas,  however, 
we  know  to  be  equal  to  pdV  (i.e.  =  RT  for  each  mole), 
hence  we  obtain 

dE=dU+pdV, 

which  is  the  analytical  expression  of  the  first  law  of  ener- 
getics. 

Starting  with  this  expression  of  the  first  law  it  is  pos- 
sible to  derive  a  number  of  mathematical  relations,  which, 
although  important  for  general  purposes,  are  not  abso- 
lutely essential  for  the  object  we  have  in  view.*  Two 
relations  which  are  thus  to  be  found  will  be  valuable 
to  us,  however,  so  that  we  shall  discuss  them  as  though 
they  were  empirical  facts,  rather  than  the  result  of  mathe- 
matical reasoning. 

It  will  be  remembered  (p.  18)  that  we  found  the  rela- 
tion between  the  pressure  and  volume  of  a  gas  under 
the  condition  that  the  temperature  remains  constant.  But 
if  we  compress  a  gas  rapidly  its  temperature  rises;  and 
if  we  allow  it  to  expand  rapidly  its  temperature  falls. 
Naturally,  Boyle's  law  holds  for  the  gas  in  these  two 
cases,  after  the  temperature,  in  consequence  of  radiation,  has 
attained  its  initial  value;  but  what  is  the  relation  between 
pressure  and  volume  immediately  after  compression  or 
expansion,  i.e.,  before  the  temperature  has  become 
equalized?  This  question  could  still  be  answered  by 
Boyle's  law  if  we  knew  the  temperature  produced  by 
the  compression  or  expansion;  but  we  do  not.  As  the 
result  of  mathematical  reasoning,  however,  this  rela- 
tion is  found  to  be  the  following: 

pl:p2::v2k:vik, 

*  See  Morgan's  Elements  of  Physical  Chemistry,  3d  ed.,  1905,  pp.  38-48. 


HEAT  AND  ITS   TRANSFORMATION.  39 


where  k  =  —,  i.e.  is  the  ratio  of  the  specific  heat  of  the 

C  V 

gas  at  constant  pressure  to  that  at  constant  volume.  When 
the  temperature  of  the  gas  is  allowed  to  change,  as  a 
result  of  the  compression  or  expansion,  then,  we  find  the 
pressures  inversely  proportional  to  the  k  powers  of  tne 
volume,  instead  of  to  the  volumes  themselves  (Boyle's 
law),  as  is  observed  when  the  change  takes  place  so 
slowly  that  the  heat  lost  or  gained  by  radiation  retains 
the  temperature  constant,  or  when  the  original  tempera- 
ture is  once  more  regained.* 

The  value  of  this  same  term  k  T  f  =— j  for  any  gas  has 
also  been  found  empirically  to  vary  with  the  number  of 

pV      T 
*  By  a  simple  transformation,  since   — =-=  — ,  where  p,   V,  and  T 

refer  to  an  isothermal,  i.e.  a  constant  temperature  (slow  change),  and 
pi,  Vi,  and  r,  refer  to  an  adiabatic,  i.e.  a  varying  temperature  (rapid 
change),  we  also  find  the  following  relations: 

rp  /-IT  \     fe _ 

—7=  I  — |          (when  the  pressure  is  retained  constant), 

and 

k-i 


.      .  (when  the  volume  is  retained  constant). 

*i      \PiJ 

By  aid  of  these  two  equations  we  can  calculate  the  temperature 
produced  by  an  adiabatic  change  of  either  pressure  or  volume.  (See 
"  Elements,"  pp.  47,  48.) 

t  k,  in  itself,  is  a  physical  constant,  for  it  can  be  determined  without 
any  knowledge  of  the  component  terms  cp  and  c?,  as  well  as  from 
the  ratio  of  these.  In  fact,  the  direct  determination  of  k  from  the 
velocity  of  sound  in  the  gas,  as  measured  in  a  Kundt  tube,  or  by  applica- 
tion of  the  formula  (  — )  =  (— )  (method  of  Clement  and  Desormes, 


"Elements,"  pp.  51,  52),  is  probably  more  accurate  than  the  indirect 
method. 


40  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

combining  weights  present  in  the  formula  weight,  although 
we  do  not  know  why  such  a  relation  should  exist,  nor 
even  the  exact  form  of  the  relation  which  does  exist. 
For  example,  it  has  been  found  that  all  substances  which 
are  "monatomic"  in  the  gaseous  state,  i.e.  all  substances 
whose  formula  weight  according  to  the  definition  is  identi- 
cal with  the  combining  weight,  Hg,  etc.,  give  a  value  of  k 
equal  approximately  to  1.67 ;  and  that  those  with  two  com- 
bining weights  to  the  formula  weight,  O2,  H2,  etc.,  lead 
to  the  value  1.4.  This  relation,  purely  empirical  as  it  is, 
is  of  very  great  value  in  those  cases  where  no  combina- 
tion of  the  gas  with  other  elements  is  known,  and  only 
the  formula  weight,  from  the  density,  can  be  determined. 
Thus  for  argon  the  formula  weight  is  found  to  be  40, 
and  although  it  is  impossible  to  find  the  combining 
weight  directly,  since  no  combinations  with  argon  have 
been  found  to  exist,  we  know  at  once  that  it  must  be  40, 
for  by  experiment  the  value  of  k  is  shown  to  be  1.67. 
The  ratio  for  "  triatomic  "  gases  is  not  so  well-established, 
but,  since  they  are  apparently  less  common,  this  detracts 
but  little  from  the  value  of  the  relation. 

One  very  important,  absolutely  general  result  of  math- 
ematical reasoning  can  be  derived  directly  from  our 
definition  of  formula  weight,  together  with  other  well- 
established  principles,  so  that  we  shall  be  able  to  consider 
it  more  in  detail  than  the  others  discussed  above.  Since 
one  formula  weight  (by  definition)  always  occupies  the 
normal  volume  under  standard  conditions,  and  since  all 
gases  expand  1/2?s  of  their  volume  at  o°  C.  for  an  increase 
of  temperature  of  i°  C.  (independent  of  the  actual  tempera- 
ture of  the  gas),  the  specific  heat  of  every  gas  at  constant 
pressure,  multiplied  by  its  formula  weight,  must  always 
exceed  its  specific  heat  at  constant  volume,  multiplied 


HEAT  AND  ITS  TRANSFORMATION.  4* 

by  the  formula  weight,  by  a  definite,  constant  value,  i.e., 
a  value  equal  to  -  =  R  units.     Since  .£  =  84800  (p.  21) 

and  the  mechanical  equivalent  of  heat  (p.  36)  is  42,600 
we  have  the  relation 


cals., 


where,  M  is  the  formula  (molecular)  weight,  and  CP  and 
Cv  are  the  formula  (molecular)  specific  heats.  The 
formula  specific  heat  at  constant  pressure,  then,  is  always 
greater  by  2  calories  than  the  formula  specific  heat  at  con- 
stant volume* 

This  relation  CP  —  CV=2  cal.  again  emphasizes  the 
difference  in  point  of  view  of  physics  and  physical  chem- 
istry (p.  20),  and  is  very  valuable  in  the  various  calculations 
with  specific  heats,  for,  when  we  know  cp  (or  cj  for  a 
gas,  and  its  formula  weight,  for  example,  we  can  readily 

find  the  value  of  cv  (or  cp).      And,  knowing  k(  =  —  ,  see 


foot-note,  p.  39)  and  the  formula  weight,  i.e.,  having  no 
knowledge  of  the  value  of  either  cp  or  cv,  we  can  calculate 

these  values.     Thus,  for  argon,  we  know  that  k=—  =  1.67, 

Cv 

and  that  40^—40^=2,  hence  £,,=0.075  and  ^=0.124. 
A  case  which  is  similar  to  the  dependence  of  specific 
heat  upon  the  volume  relations  is  the  dependence  of  the 
heat  evolved  by  a  chemical  reaction  upon  the  volume 
relations  of  the  system.  In  other  words,  certain  reactions 
are  observed  to  evolve  a  different  amount  of  heat  when 

*  For  a  table  of  these  values,  see  Elements,  p.  49. 


42  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

the  volume  of  the  system  is  allowed  to  increase  than  when 
it  is  retained  constant.  Naturally,  in  such  cases,  the  prog- 
ress of  the  reaction  itself  must  involve  a  change  in  volume, 
for  it  has  been  observed  that  a  reaction  is  unaffected  by 
changes  in  volume  when  the  initial  and  final  substances, 
under  like  conditions,  occupy  the  same  volume,  i.e., 
the  reaction  is  unaffected  unless  the  change  in  volume 
which  we  bring  about  affects  the  two  sides  differently. 
Thus  the  volume  of  the  system  has  no  influence  whatever 
upon  the  direction  of,  or  the  heat  evolved  by,  the  gaseous 
reaction 


for  we  have  two  formula  weights  of  substance  on  each  side 
and  hence  (by  definition)  the  volume  of  each  side,  under 
like  conditions,  must  be  the  same  and  influenced  equally. 
A  general  rule  in  this  connection  as  to  the  effect  of  a 
change  of  volume  upon  the  direction  (not  the  heat  evolved) 
of  a  reaction  is  the  so-called  principle  of  Le  Chatelier, 
which  may  be  expressed  as  follows:  Any  change  from  the 
exterior  in  the  factors  governing  the  reaction  is  always 
followed  by  a  reverse  change  within  the  system  itself.  Thus 
in  the  reaction 


for  example,  decreased  volume  will  cause  the  reaction 
to  go  toward  the  right  to  a  smaller  degree  than  before, 
i.e.,  since,  in  going  toward  the  right,  an  increase  in  volume 
is  involved,  decreased  volume  favors  the  reaction  going  in 
the  reverse  direction  and  involving  a  decrease  in  volume. 
Although  this  shows  the  effect  of  external  conditions 


HEAT  AND  ITS   TRANSFORMATION.  43 

upon  the  direction  of  a  reaction,  it  does  not  give  us  any 
idea  of  the  effect  upon  the  heat  evolved,  except  that  it 
must  necessarily  be  different,  because  the  reaction  takes 
place  to  a  smaller  extent,  owing  to  the  decreased  volume. 
The  point  alluded  to  above,  however,  was  the  effect  of 
constant  volume  or  constant  pressure,  when  the  reaction 
still  continues  to  take  place  normally  and  to  its  jull  extent. 
Here,  as  will  be  seen  immediately,  it  is  only  a  question  of  the 
loss  of  heat  due  to  external  work  performed  in  expanding, 
or  of  the  absorption  of  heat  due  to  the  work  done  upon  it 
when  the  volume  decreases.  Starting  with  the  reaction 

=2H2O  4-  2X67484  cal. 


Gaseous.         Liquid. 

at  1  8°  and  constant  volume,  for  example,  we  can  readily 
calculate  the  value  under  constant  pressure,  for  3  moles  of 
substance  in  the  gaseous  state  are  transformed  by  the 
reaction  into  36  grams  (about  36  c.c.)  of  liquid  water. 
At  constant  pressure,  that  of  the  atmosphere,  for  example, 

the  volume  of  the  system  would  decrease  from  3  X  22.4-^ 

liters  to  36  c.c.,  and  heat,  equivalent  to  the  work  done 
upon  the  system  during  this  reduction  of  volume,  will  be 
absorbed.  In  case  the  reaction  took  place  in  the  opposite 
direction,  i.e.,  liquid  water  were  decomposed  into  gaseous 
oxygen  and  hydrogen,  this  same  amount  of  heat  would 
be  lost  under  constant  pressure,  for  the  system  would 
have  to  perform  work  in  expanding.  Since  the  work 
in  either  case  would  be  equal  to  the  product  of  pressure 
and  volume,  i.e.  equal  to  RT  or  2  Teal,  for  each  mole 
of  gas,  the  difference  in  heat  evolved  in  the  two  cases 
would  be  3  X  2  T  calories.  And  this  work  will  depend 
in  value  only  upon  the  temperature,  and  will  be  inde- 


44  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

pendent  of  the  pressure  or  volume  of  the  gas  (pp.  29  and 
30).     Expressing  this  mathematically,  then,  we  have 


or 


where  n  is  the  number  of  moles  of  gas  formed  or  ab- 
sorbed. Qp  will  always  be  larger  than  Qv,  then,  when  a 
gas  is  absorbed  by  the  reaction,  i.e.,  when  the  volume 
decreases;  otherwise  Qv  will  be  the  larger. 

The  second  law  of  thermodynamics.  —  Thus  far,  in  con- 
sidering the  first  law  of  thermodynamics  or  energetics, 
we  have  only  made  use  of  the  principle,  based  upon 
experience,  that  when  a  quantity  of  energy  disappears 
at  any  place,  a  precisely  equal  quantity  (in  the  same 
terms)  of  energy  appears  simultaneously  elsewhere. 
And  this  condition  was  fulfilled  in  all  the  things  we  dis- 
cussed. We  must  now  consider  another,  also  empirical, 
principle  relating  to  energy  and  its  transformation,  the 
so-called  second  law  of  thermodynamics.  This  principle 
may  be  expressed  in  the  form,  a  perpetual  motion  of  the 
second  kind  is  impossible,  i.e.,  heat  cannat  be  made  to  go 
from  a  colder  to  a  warmer  body  without  the  expenditure 
of  work. 

As  has  already  been  mentioned  (p.  13),  the  transfer 
of  heat  is  conditioned  by  the  temperature,  i.e.,  the  inten- 
sity factor  of  heat  energy.  But  the  simple  transfer  of 
heat  does  not  always  result  in  the  appearance  of  mechan- 
ical work.  Here,  consequently,  we  must  consider  the 
questions  as  to  how  heat  is  to  be  transferred  that  work 
may  result  from  the  transference,  and  how  the  amount 
of  work  resulting  depends  upon  the  amount  of  heat 


HEAT  AND  ITS   TRANSFORMATION.  45 

transferred — for  we  know  that  although  all  forms  of 
energy  may  be  completely  transformed  into  heat,  the 
reverse  transformation  is  never  complete.  It  will  be 
seen,  then,  that  we  are  only  to  discuss  the  second  law  of 
thermodynamics  in  order  that  we  may  obtain  a  general 
and  absolutely  essential  rule  for  our  later  work,  and  not 
from  the  standpoint  of  pure  thermodynamics. 

The  transformation  of  heat  into  work  takes  place 
only  through  the  medium  of  a  gaseous  body.  Thus  by 
absorbing  an  amount  of  heat  a  gas  expands,  performing 
external  work,  until  its  temperature,  which  must  be 
higher  than  that  of  the  environment,  is  reduced  to  this. 
By  this  process  heat  at  a  high  temperature  is  absorbed 
by  the  gas  and  is  partly  transformed  into  mechanical 
work,  i.e.,  until  the  temperature  of  the  gas  has  fallen 
to  that  of  the  environment.  No  further  transformation 
of  that  amount  of  heat  is  possible,  then,  for  heat  cannot 
go  from  one  body  to  another  (second  law)  when  both 
are  at  the  same  temperature. 

The  relation  between  the  amount  of  heat  absorbed 
in  any  process  and  the  consequent  maximum  amount 
of  mechanical  work  resulting  can  be  derived  by  aid  of 
the  following  cycle,  composed  of  assumed,  ideal  processes : 

i.  Assume  an  ideal  gas,  enclosed  in  a  cylinder  with 
a  movable  piston,  at  a  certain  temperature  and  pressure. 
Imagine  the  cylinder  to  be  placed  upon  a  heating-bath 
at  the  temperature  T\9  allowing  the  volume  to  increase 
at  a  constant  pressure  which  is  }ust  greater  than  that  of 
the  atmosphere.  By  the  expansion  the  gas  will  cool, 
but  so  long  as  it  remains  on  the  heating-bath  it  will 
absorb  heat,  retaining  the  temperature  constant.  If  this 
heat  absorbed  is  Qi,  the  initial  volume  is  vi,  and  its 
final  one  v^  the  temperature  being  T\>  the  work  done 


46  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 
will  be  equal  to    /    pdv,   and  expressing  both  in  the 

t/Vi 

rT 
same  units  we  have,  since  p= —  (p.  20), 


dQ^rT— 

v 


or 


^  Jlie  gas  is  next  allowed  to  expand  adiabatically 
away  from  the  heating  bath  until  the  temperature  falls 
to  T2.  For  this,  the  new  volume  being  1*3,  we  have  the 
relation  (p.  39) 


*-: 


3.  Next  the  pressure  is  increased  until  the  volume 
is  decreased  to  v^  heat  to  the  amount  Q2  being  removed, 
so  that  the  temperature  remains  constant  at  TV  The 

work  done  here  by  the  gas  is  -  [pdv,  and  we  have 

t/v» 


Q.-iT.log.g. 

4.  Finally  the  gas  is  compressed  adiabatically  until  the 
original  volume  Vi,  and  the  original  temperature  J"i,  are 
reached.  For  this  we  have  the  relation 


T2 
Ti 


HEAT  AND  ITS   TRANSFORMATION.  47 

We  have  thus  carried  the  gas  through  a  series  of  ideal 
changes,  and  have  finally  the  same  conditions  as  those 
with  which  we  started.  The  amount  of  heat  Qi  has 
been  absorbed  at  the  higher  temperature  TI,  and  a  smaller 
amount  Q2  has  been  evolved  at  a  lower  temperature  T2, 
and  the  rest  has  been  transformed  into  work;  i.e.,  Q, 
the  amount  of  work  produced  (in  terms  of  heat),  is  equal 
to  Qi—  (?2>  for  the  amount  of  heat  Q2  has  simply  been 
transferred  from  TI  to  T2. 

The  relation  between  Q\  and  Q2,  then,  is  obviously 
the  following: 


but 


hence 


*>2      ^3        .          ,         V2      ,          #3 

---,    ..e.,log.-=log,-; 


i.e.,  the  amounts  of  heat  absorbed  and  liberated  are  pro- 
portional to  the  absolute  temperatures  of  the  processes. 

Since  Q  =  Qi  —  Q2  is  the  heat  transformed  into  work, 
we  have,  then, 


Qi-Q*    Tt-T* 

/^  T1  > 


and 


48  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

The  heat  transformed  into  work  by  any  reversible  process 
is  to  that  transferred  from  the  higher  to  the  tower  tem- 
perature as  the  difference  in  temperature  is  to  the  lower 
absolute  temperature.  Or,  the  heat  transformed  into  work  is 
to  that  absorbed  as  the  difference  in  temperature  is  to  the 
higher  absolute  temperature.  Or,  from  the  other  stand- 
point, the  work  in  calories  necessary  to  transfer  a  certain 
amount  of  heat  from  one  temperature  to  a  higher  one  by  a 
reversible  process  is  to  the  amount  of  heat  as  the  temperature 
interval  is  to  the  final  high  absolute  temperature* 

It  will  be  observed  from  the  above  that  it  is  only  at 
the  absolute  zero  that  all  heat  will  be  transformed  into 
work,  for  when  T2  =  o,  Q2  will  be  zero,  and  Qi-Q2  =  Q\- 

In  its  second  form  this  general  rule  is  that  which  is  used 
in  engineering  work  to  find  the  efficiency  of  heat-engines. 
Thus,  at  best,  an  engine  working  with  a  boiler  tem- 
perature of  200°  C.,  the  temperature  of  the  condenser 
being  50°  C.,  will  have  an  efficiency  of  15%73  =  o.3i7 

'-^= — -  j,  i.e.,  assuming  no  heat  to  be  lost  either  by 

radiation  or  by  the  work  used  in  overcoming  friction, 
such  an  engine  would  transform  31.7%  of  the  heat 
absorbed  into  work. 

It  will  be  observed  that  in  general  we  may  write  this 
relation  in  the  form 


From  this  form  it  is  possible  for  us  to  find  the  capacity 
factor  of  heat  energy,  for  —  J  T  expresses  the  variation  of 

*  The  reader  who  does  not  follow  the  mathematical  reasoning  is 
advised  at  least  to  become  thoroughly  familiar  with  the  laws  in  italics,  as 
well  as  the  three  last  formulas,  for  they  are  vital  in  some  of  our  later  work. 


HEAT  AND  ITS  TRANSFORMATION.  49 

heat  energy,  i.e.,  shows  the  amount  of  heat  transformed 
into  work  when  the  heat  Q  has  its  temperature  reduced 
AT0.  Since  the  temperature  is  the  intensity  factor  of 
heat  energy  (p.  13),  the  capacity  factor  must  obviously 

be  the  quantity  7p.    This  factor  is  called  the  entropy,  and 

is  very  important  in  rinding  when  a  reaction  of  any 
kind  will  take  place. 

By  the  first  law  of  thermodynamics  we  find  how  a 
reaction  will  take  place,  but  have  no  means  of  deciding 
whether  or  not  it  will  take  place.  By  the  second  law 
we  find  that  only  that  reaction  can  take  place  of  itself  by 
which  the  entropy  of  the  system  will  increase,  for  the 

denominator  of  the  expression  •=  must  always  decrease 

if  a  reaction  is  to  take  place  of  itself,  i.e.,  heat  can  only 
go  from  a  warmer  to  a  cooler  body  of  itself. 

It  is  in  only  the  third  form  (p.  48)  that  we  shall  use 
the  second  law  of  thermodynamics  for  the  derivation  of 
formulas,  but  too  much  stress  cannot  be  laid  upon  the 
capacity  factor  of  heat  energy,  the  entropy,  which  we 
have  deduced  from  the  second  law,  for  wherever  we 
have  to  consider  heat  energy  we  shall  have  to  employ 
its  factors. 


CHAPTER  IV. 

SOLUTIONS. 

The  formula  (molecular)  weight  in  the  liquid  and 
solid  states. — Most  of  that  which  we  have  found  to  hold 
true  for  gases  is  directly  applicable  to  substances  in  the 
state  of  solution;  and  even  a  cursory  glance  at  the  con- 
tents of  this  chapter  will  convince  one  of  it,  and  justify 
the  time  we  have  devoted  to  the  discussion  of  gaseous 
systems — all  unnecessary,  as  this  may  have  appeared 
at  first  sight.  Before  discussing  this  similarity  of  behavior, 
however,  it  would  first  seem  necessary  to  consider  some- 
thing of  the  general  relations  of  the  other  two  states 
(liquid  and  solid)  in  which  the  substances  composing 
the  solution  may  have  previously  existed,  i.e.,  of  the 
components  of  the  system.  But  experience  shows  that 
such  a  knowledge  is  not  essential,  for  much  more  is 
known  of  the  laws  governing  the  behavior  of  solutions 
than  of  those  regulating  that  of  either  liquids  or  solids. 
In  fact,  the  behavior  of  a  solution  differs  so  utterly  from 
that  of  the  pure,  liquid  solvent,  that  it  may  be  followed 
without  any  further  knowledge  of  the  behavior  of  the 
solvent  than  that  comprised  in  the  few  and  well-known 
physical  facts  with  which  we  assume  familiarity.  The 
only  purely  chemical  relation  to  be  considered,  indeed, 
for  either  liquids  or  solids,  is  the  definition  of  the  formula 
weight,  and  this  is  only  necessary  that  we  may  be  able 

5° 


SOLUTIONS.  51 

to  follow  tne  cnanges  in  the  formula  weight  as  the  sub- 
stance passes  through  the  various  possible  phases  of 
its  existence,  i.e.,  the  gaseous,  liquid,  solid,  and  dissolved 
states. 

The  formula  weight  in  the  gaseous  state,  as  we  have 
already  observed,  may  be  denned  hi  several  ways;  and 
all  definitions  will  lead  to  the  same  result,  for  all  are 
based  upon  the  same  fundamental  fact,  although  ex- 
pressed in  other  terms.  Thus  we  may  say,  the  formula 
weight  of  a  substance  in  the  gaseous  state  is  that  weight 
which  will  occupy,  under  like  conditions,  the  same  volume 
as  32  units  of  weight  oj  oxygen:  or,  is  the  weight  which 
when  multiplied  by  the  difference  in  the  specific  heats 
under  constant  pressure  and  at  constant  volume  (i.e., 
cp-c^)  will  give  the  value  2  calories;  or,  is  the  weight 
oj  the  gas,  occupying  any  volume  under  any  pressure, 
which  will  do  the  external  work  RT  when  its  tempera- 
ture is  raised  i°  C.,  i.e.,  from  T—i  to  T.  And  still 
other  definitions  could  be  given.  In  short,  then,  the 
formula  weight  in  the  gaseous  state  is  a  very  definite 
conception,  and  can  be  readily  determined  experi- 
mentally. 

In  the  liquid  state,*  on  the  other  hand,  we  have  only 
one  general  definition  for  the  formula  weight,  and  this 
depends  upon  the  so-called  surface  tension  of  the  liquid, 
i.e.,  upon  the  force  in  dynes  necessary  to  form  a  liquid 
surface  with  an  area  of  i  square  centimeter.  The  sur- 
face tension  of  a  liquid  can  be  found  indirectly  from 
the  height  to  which  the  liquid  ascends  in  a  capillary 
tube  of  known  radius,  and  is  equal  to  one-half  the  product 


*  For  further  information  as  to  the  liquid  state,  see  "The  Elements 
of  Physical  Chemistry,"  3d  ed.,  1905,  pp.  61-90. 


52  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

of  the  height  (in  centimeters)  into  the  radius  (also  in 
centimeters)  into  the  specific  gravity,  i.e.,  the  surface 
tension  =  l/2  hrs.  According  to  this  definition,  the 
formula  weight  of  a  liquid  is  that  weight  in  grams  which 
gives  such  a  surface  that  its  increase,  due  to  a  heating 
of  1°  C.,  involves  the  surface  work  of  2.12  ergs;  in  other 
words,  the  temperature  coefficient  of  the  formula  (molec- 
ular) surface  tension  (i.e.,  the  temperature  coefficient  of 
the  tension  of  the  surface  of  i  mole)  is  always  the  same, 
independent  of  the  nature  of  the  substance.  This  defi- 
nition, it  will  be  observed,  is  very  similar  to  one  of  those 
given  above  for  gases,  except  that  surface  energy  is 
involved  in  place  of  volume  energy. 

There  is  one  question  which  may  arise  here,  which, 
if  not  answered,  can  lead  to  confusion;  it  is,  How  was 
this  definition  obtained  originally?  Naturally,  we  may 
not  go  very  deeply  into  this  matter  here,  but  at  any 
rate  the  empirical  origin  of  the  definition  may  be  pointed 
out.  By  assuming  the  formula  weight  to  be  the  same 
in  the  liquid  as  in  the  gaseous  state,  the  factor  2.12  ergs 
was  found  to  remain  constant  for  a  large  number  of 
substances.  The  natural  inference,  then,  was  that 
this  value  (2.12)  is  typical  of  the  formula  weight,  and 
that  any  variation  from  it,  when  the  gaseous  formula 
weight  is  assumed,  shows  that  the  formula  weight  changes 
as  the  substance  goes  from  the  gaseous  to  the  liquid 
state.  Using  this  same  plan,  other  methods  decidedly 
more  restricted  in  their  scope  have  been  developed,  and 
as  the  same  discrepancy  for  any  one  substance  is  shown 
by  all,  our  conclusion  is  considered  to  be  justified. 

It  must  be  acknowledged  that  our  definition  of  for- 
mula weight  in  the  liquid  state  is  very  much  less  sat- 
isfactory than  that  for  the  gaseous  state,  but  unfortu- 


SOLUTIONS.  53 

nately  we  find  the  relations  for  the  solid  state  *  still  less 
satisfactory,  for  as  yet  it  has  been  impossible  to  find  any 
definition  for  the  formula  weight  in  the  solid  state.  It 
is  true  that  at  first  glance  it  would  seem  that  the  law 
of  Dulong  and  Petit — that  the  combining  (atomic) 
weight  of  an  element  (excepting  carbon,  boron,  and 
silicon)  when  multiplied  by  its  specific  heat  always  gives 
the  approximately  constant  value  6.34 — would  suffice 
for  this  purpose,  since  the  sum  of  the  combining  weights, 
i.e.  the  formula  weight,  multiplied  by  the  specific  heat 
of  the  compound  should  give  approximately  the  same 
value  as  is  obtained  when  6.34  is  multiplied  by  the  num- 
ber of  combining  weights  in  the  formula  weight.  But 
a  moment's  thought  shows  that  this  relation  is  only  of 
value  in  that  it  insures  uniformity  in  the  choice  of  the 
combining  ratio  to  be  used  as  the  combining  (atomic) 
weight;  in  other  words,  at  best,  it  is  only  another  method 
for  fixing  the  combining  (atomic)  weight  of  an  element 
without  the  necessity  of  knowing  or  studying  the  com- 
pounds which  it  may  form  (see  pages  4  and  40),  and 
has  nothing  to  do  with  fixing  the  formula  weight  of  an 
elementary  compound.  Thus  analysis  shows  the  exist- 
ence of  two  chlorides  of  mercury,  the  simplest  formulas, 
using  at  least  one  combining  weight  of  each  element,  being 
HgCl  and  HgCl2,  but  the  application  of  the  law  of  Dulong 
and  Petit  does  not  enable  us  to  state  whether  the  formula 
weight  in  the  solid  state  is  HgCl  or  Hg2Cl2,t  etc.,  any 
more  than  the  analytical  result  does.  And  the  same 
is  true  for  the  other  chloride.  It  is  only  because  the 


*  For  further  information  on  this  state,  see  "  Elements,"  pp.  91-103. 
t  For   HgCl  we    have    (200+35.5)0.052=12.25;    for    Hg2Cl2  (400 
+  71)0.052=24.5;  while  2X6.34=12.68,  and  4X6.34  =  25.36. 


54  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

formulas  Hg2Cl2  and  HgCl2  (and  not  H^CU)  are  found 
under  certain  conditions  in  the  gaseous  state  that  they 
are  assumed  in  the  solid  state.  In  all, reactions  between 
solids,  then,  it  is  only  the  actual  weight  of  the  reacting 
substances  which  is  essential,  and  the  very  fact  that  we 
cannot  define  formula  weight  in  the  solid  state  shows 
its  utter  lack  of  chemical  or  physical  influence.  If  at 
any  time  this  should  be  changed,  and  it  appear  that  the 
formula  weight  does  exert  an  influence  experimentally, 
naturally  all  our  difficulty  would  disappear,  for  then  a 
definition  could  at  once  be  derived  from  observations 
of  this  influence. 

The  molecular  weight  of  a  dissolved  substance,  in 
contrast  to  that  of  a  liquid  or  a  solid,  plays  a  very  impor- 
tant role  indeed,  and  is  just  as  definite  in  its  meaning  as 
that  in  the  gaseous  state.  Indeed,  it  will  almost  appear 
to  the  reader,  after  his  perusal  of  the  following  pages, 
that  the  formula  weight  is  the  significant  and  fundamental 
conception  in  the  consideration  of  dissolved  substances; 
and  certainly  its  paramount  practical  importance  is 
decidedly  striking,  especially  when  contrasted  with  the 
slight  importance  of,  and  impossibility  of  defining  chem- 
ically, the  formula  weight  in  the  liquid  state,  and  its 
apparent  utter  lack  of  either  chemical  or  physical  influ- 
ence in  the  solid  state. 

Osmotic  pressure. — It  is  a  well-known  fact  that,  when 
a  solution  is  carefully  superimposed  upon  another  con- 
taining a  different  amount  of  the  substance,  diffusion 
takes  place,  and  that  this  diffusion  is  so  directed  that  the 
entire  system  finally  becomes  homogeneous.  There  is, 
then,  a  tendency  for  the  substance  in  solution  to  become 
uniformly  distributed  throughout  the  volume  of  liquid 
accessible  to  it.  Following  the  plan  we  have  adopted, 


SOLUTIONS.  55 

i.e.  considering  only  the  facts  and  avoiding  all  hypothesis, 
we  shall  study  the  facts  of  this  phenomenon,  and  the 
general  relations  to  be  derived  from  them,  without  at 
the  same  time  trying  to  picture  their  cause,  or  to  explain 
their  origin,  by  hypothesis.  • 

The  facts  of  diffusion  may  be  summed  up  by  the 
following  statement:  A  system  composed  of  a  substance 
dissolving  in  a  solvent,  or  of  two  solutions  of  different 
strength,  one  being  superimposed  upon  the  other,  behaves 
as  though  there  existed  an  attraction  between  the  sub- 
stance and  the  solvent,  which  ultimately  produces  a 
uniform  concentration  of  the  dissolved  substance  through- 
out the  entire  system.  It  is  to  be  noted  here  that  this 
is  not  in  any  way  an  assumption,  but  simply  a  description 
of  the  behavior  of  the  system  as  we  observe  it. 

The  difference  in  concentration  of  the  two  liquids  in 
contact  has  been  observed  to  influence  very  largely  the 
force  of  diffusion  and  in  such  a  way  that  the  greater  the 
concentration-difference  the  greater  will  be  the  force  of 
diffusion.  To  find  the  quantitive  relation  existing  be- 
tween the  amount  of  dissolved  substance  in  the  solution 
and  the  force  of  diffusion  into  the  pure  solvent,  however, 
it  is  necessary  for  us  to  find  a  method  of  measuring 
the  force  of  diffusion.  Naturally,  such  a  measurement 
would  not  be  difficult  if  we  could  obtain  a  partition, 
to  be  placed  between  the  solution  and  the  pure  solvent, 
which  would  allow  passage  to  the  solvent  and  not  to 
the  solute,  for  then  we  could  simply  measure  the  force 
with  which  the  pure  solvent  goes  through  the  partition. 
At  first  glance  the  difficulty  of  obtaining  such  a  partition, 
i.e.  one  which  in  the  case  of  water  and  an  aqueous  sugar 
solution,  for  example,  will  allow  water,  but  not  the  dis- 
solved sugar,  to  pass  through,  seems  absolutely  insur- 


$6  PHYSICAL  CHEMISTRY  FOR.  ELECTRICAL  ENGINEERS. 

mountable.  It  is  not  so,  however,  for  such  semi  permeable 
films,  as  they  are  called,  are  very  widely  distributed  in 
nature,  and  can  also  be  readily  prepared  by  artificial 
means.  In  the  first  class,  for  instance,  we  find  the  pro- 
toplasm of  which  living  organisms  are  made  up,  while 
of  the  second,  a  good  example  is  a  film  of  copper  ferro- 
cyanide,  as  formed  by  the  reaction  of  copper  sulphate 
and  potassium  ferrocyanide. 


By  aid  of  a  copper  ferrocyanide  film,  permeable  to  water 
but  not  to  sugar  and  many  other  substances,  Pfeffer 
was  able  to  measure  the  apparent  attraction  between 
substance  in  solution  and  the  pure  solvent,  the  osmotic 
pressure,  as  it  is  called.  The  principle  of  Pfeffer's  appa- 
ratus is  illustrated  by  the  figure  above.  The  cylinder  A, 
closed  at  the  bottom,  is  of  porous  clay  and  is  intended  to 
support  the  semipermeable  film.  This  film  is  prepared 
by  filling  the  porous  cup  with  a  solution  of  potassium 
ferrocyanide  and  allowing  it  to  stand  for  a  day  in  a  solution 
of  copper  sulphate.  In  this  way  the  pores  of  the  cup  are 
filled  with  the  precipitated  copper  ferrocyanide,  which, 
although  permeable  to  water,  is  not  so  to  the  dissolved 
substance. 


SOLUTIONS.  57 

To  make  a  measurement  with  this  apparatus  the  cup 
so  prepared  is  filled  with  the  solution  to  be  studied 
and  the  rubber  stopper  CC  inserted  in  such  a  way  that 
the  solution  rises  a  short  distance  in  the  measuring-tube. 
The  cell  is  next  immersed  in  water  and  retained  in  position 
by  the  cork  BB.  The  liquid  is  riow  observed  to  rise  very 
slowly  in  the  tube  until  equilibrium  is  finally  attained, 
i.e.,  until  the  weight  of  the  liquid  in  the  tube  just  counter- 
acts the  pressure  with  which  the  water  enters  the  cell. 
.Since  the  entrance  of  water  into  the  cell  decreases  the 
concentration  of  the  solution  within  it,  the  actual  measure- 
ments are  usually  made  by  aid  of  a  mercury  manometer, 
so  arranged  that  the  change  in  volume  is  negligible.  In 
this  way  the  pressure  observed  is  that  which  just  pre- 
vents the  entrance  of  water  into  the  original  solution, 
diluted  only  to  an  exceedingly  small  extent,  and  not,  as 
in  the  above  description,  into  one  diluted  by  an  amount 
of  water  equal  in  volume  to  the  liquid  which  rises  in  the 
tube.* 

From  the  experiment,  then,  we  know  that  a  certain 
definite  pressure  is  necessary  to  prevent  water  going 
into  the  porous  cell  to  dilute  the  solution  contained 
in  it.  This  is  what  we  shall  call  osmotic  pressure.  When- 
ever we  use  this  term,  consequently,  it  is  without  any 
assumption  as  to  the  cause  of  the  pressure,  and  is  merely 
expressive  of  the  experimental  fact  that  it  is  necessary 
to  exert  a  pressure  in  order  to  prevent  pure  solvent  flow- 
ing through  a  semipermeable  film  to  dilute  the  solution, 
which  is  surrounded  by  it. 

It  will  be  noted  here  that  we  do  not  consider  what 

*  For  experimental  details  of  very  accurate  measurements,  where  the 
pressures  rise  as  high  as  25  atmospheres,  see  Morse  and  Frazer,  Am. 
Chem.  Jour.,  34,  i,  July  1905. 


58  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

osmotic  pressure  really  is,  but  what  we  mean  by  the 
word  osmotic  pressure.  This  again  emphasizes  our 
standpoint.  By  defining  each  concept  in  terms  of  experi- 
ment, and  studying  the  facts  of  the  phenomenon,  asking 
only  how  it  takes  place  and  never  why,  we  may  always 
be  certain  that  our  knowledge  is  based  upon  facts  alone, 
and  is  perfectly  free  from  any  trace  of  hypothesis. 

Pfeffer's  results  showed  that  this  osmotic  pressure 
increases  with  the  amount  of  substance  dissolved,  and 
for  any  one  concentration  is  proportional  to  the  abso- 
lute temperature  of  the  solution.  The  numerical  values 
for  various  sugar  solutions  at  15°  C.,  as  found  by  Pfeffer, 
are  given  below,  c  being  the  percentage  of  sugar,  and 
p  the  pressure,  in  centimeters  of  a  column  of  mercury, 
which  will  just  preserve  equilibrium  between  the  water 
and  the  solution,  i.e.  will  just  prevent  water  flowing 
through  the  semipermeable  film. 

p/c 


53-5 


The  principal  difficulty  experienced  by  Pfeffer  was 
the  breaking  down  of  the  film  under  the  pressure  exerted, 
which,  naturally,  allowed  the  solution  enclosed  to  escape 
and  so  led  to  a  smaller  result  than  would  otherwise 
have  been  obtained.  Such  an  action  can  always  be 
detected,  however,  by  testing  the  pure  solvent  for  sugar; 
and,  indeed,  the  presence  of  this  was  always  observed 
when  the  stronger  solutions  were  measured,  and  was 
the  reason  why  the  6%  concentration  was  the  highest 
employed. 

From    Pfeffer's    observations,    as   given    above,    it   is 


c 

P 

p/c 

c 

P 

I 

53-8 

53-8 

4 

208.2 

I 

53-2 

53-2 

6 

30-75 

2 

101.6 

50.8 

i 

53-5 

2-74 

151.8 

55-4 

SOLUTIONS.  59 

quite  evident  that,  within  the  experimental  error,  there 
exists  a  general  relation  between  concentration  and 
osmotic  pressure,  for  the  term  p/c  is  practically  con- 
stant, at  constant  temperature.  But  c,  the  number 
of  grams  of  solute  in  100  grams  of  solution,  is  obviously 
the  reciprocal  of  the  volume  of  solution,  v,  containing 

i  gram  of  solute,  i.e.,  c=  — .     We  have,  then,  however, 
in  place  of  p/c  =  constant,  the  relation 
pv=  constant, 

and  further,  since  p  is  proportional  to  the  absolute  tem- 
perature, T,  i.e.,  p=o,  when  r=o,  we  may  write 

pv= constant  XT. 

This  equation  is  so  strikingly  similar  to  the  one  already 
derived  for  gases  (pp.  19  and  20)  that  it  at  once  suggests 
an  analogy  between  the  behavior  *  of  a  substance  in  solu- 
tion and  one  in  the  gaseous  state,  and  makes  the  deter- 
mination of  the  value  of  the  constant  a  point  of  extreme 
interest.  Since  for  the  gaseous  state  we  have  found 
the  constant  R  (in  pV  =  RT)  to  be  a  constant  for  all  sub- 
stances, when  the  volume  V  is  that  occupied  by  the  for- 
mula weight  (i.e.,  the  combining  weight  of  an  element 
multiplied  by  some  small  whole  number,  or  the  sum 
of  the  combining  weights  of  the  elements  composing 
a  compound),  it  is  but  natural  to  think  that  some  such 
analogous  result  may  be  found  for  substances  in  solu 

*  It  is  to  be  remembered  here  that  experiment  shows  the  analogy  in 
behavior,  and  does  not  justify  the  assumption  that  the  reason  for  the 
behavior  is  the  same.  Although  it  has  been  possible  by  aid  oj  hypoth- 
esis to  formulate  a  kinetic  theory  (i.e.,  in  one  sense  of  the  word  a  hy- 
pothesis) of  gases,  all  efforts  to  do  likewise  with  liquids  or  solutions 
have  been  futile. 


60  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

tion.  Such  a  result,  however,  would  lead  immediately 
to  a  definition  of  formula  weight  in  solution,  if  such 
a  conception  exist,  for  it  would  show  that  the  formula 
weight  exerts  an  easily  observed  influence  upon  the 
physical  behavior  of  the  solution.  And  its  importance 
would  probably  not  be  restricted  to  osmotic  pressure 
alone,  for  presumably  then  the  physical  behavior  of  a 
solution  in  other  respects  would  also  be  influenced  by 
the  formula  weight,  and  we  could  employ  the  defini- 
tions for  formula  weight  so  obtained  not  only  as  defi- 
nitions, but,  knowing  the  formula  weight,  could  also 
calculate  the  exact  behavior  in  these  other  respects  of 
any  substancce  dissolved  in  a  solvent. 

In  order  to  find  whether  the  constant  in  our  equation 
has  any  definite  relation  to  that  for  gases,  however,  it 
is  necessary  to  calculate  it  for  one  formula  weight  in 
the  dissolved  state.  Here,  naturally,  arises  the  question 
as  to  what  the  formula  weight  in  solution  is,  for  as  yet 
we  have  not  derived  any  definition  of  it.  Let  us  assume 
temporarily  that  sugar  has  the  same  formula  weight 
in  solution  as  it  is  generally  assumed  *  to  have  in  the 

*  It  will  be  observed  after  a  survey  of  chemical  compounds  that 
many  formulas  are  employed  which  cannot  be  proven  experimentally. 
Under  the  atomic  hypothesis  it  was  usually  assumed  that  the  formula 
of  a  compound  was  that  simple  relation  which  would  give  at  least  one 
atom  of  the  substance  present  to  the  smallest  extent,  and  would  lead 
to  a  whole  number  of  atoms  of  the  other  constituents.  Or,  from  the 
hypothesis-free  standpoint,  would  involve  a  whole  number  of  combining 
weights  of  each  constituent.  This,  naturally,  is  but  following  out  the 
consequences  of  the  choice  of  hydrogen  (or  a/16  oxygen)  as  the  basis 
of  the  combining  weights.  But  in  either  case  the  assumption  of  a 
formula  weight  without  experimental  foundation  is  not  justified.  In 
the  case  of  sugar  the  structure,  the  products  of  decomposition,  etc., 
have  led  to  the  formula  Cl2H22On,  and  we  assume  without  direct  ex- 
perimental evidence  that  this  would  also  be  the  formula  in  the  gaseous 
state  if  no  decomposition  took  place; 


SOLUTIONS.  6l 


other  states,  viz.,  C^H^On,  and  calculate  the  constant 
on  the  assumption  that  the  v  in  our  equation  is  the  volume 
of  solution  in  which  this  formula  weight  (342  grams) 
is  dissolved. 

Pfeffer  found  for  a  i%  solution  of  sugar  (i.e.,  342 
grams  in  32,400  c.c.)  at  o°  an  osmotic  pressure  of 
49.3X13.6  =  671  grams  per  square  centimeter;  hence 

671X34200 
constant  =  —  -  =  84200, 


273  273 

i.e.,  assuming  that  sugar  has  the  formula  Ci2H22Oii  in 
solution  we  find  that  the  osmotic  constant  is  identical, 
within  the  experimental  error,  with  the  gaseous  constant 
(p.  21)  which  would  be  obtained  if  sugar  in  the  gaseous 
state  had  the  same  formula. 

This  same  constant  has  also  been  obtained  for  many 
other  organic  substances,  using  the  customary  formula 
weights  (p.  60)  (many  of  which  can  be  confirmed  by  aid 
of  the  gaseous  density),  and  we  may  say  in  general  that 
it  enables  us  to  find  the  formula  weight  from  the  observed 
osmotic  pressure,  or,  knowing  the  formula  weight,  to 
calculate  the  osmotic  pressure.  It  is  true  that  we  find  in 
many  cases  that  this  constant  is  only  obtained  when  the 
formula  weight  in  solution  is  taken  as  some  small 
multiple  of  that  in  the  gaseous  state,  but  in  every  such 
case  it  can  be  experimentally  shown,  by  a  method  given 
below,  that  the  formula  weight  undergoes  a  change 
when  the  substance  goes  out  of  that  solvent  in  which  its 
behavior  seems  abnormal,  into  another  in  which  it  is 
normal,  i.e.,  in  which  the  assumption  of  the  gaseous 
formula  weight  in  solution  leads  to  the  above  constant. 
Naturally,  direct  osmotic  pressure  observations  with 


62  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

solvents  other  than  water  cannot  be  made  with  the 
ferrocyanide  film,  but  it  has  been  possible  to  find  other 
substances  which  are  permeable  to  other  solvents  and  yet 
not  to  dissolved  substances.  Thus  vulcanite  and  rubber 
have  been  found  permeable  to  ether  and  not  to  alcohol, 
so  that  it  is  possible  to  measure  the  pressure  with  which 
ether  goes  through  the  film  to  dilute  a  solution  of  alcohol 
in  ether.  Pressures  of  this  kind  have  been  observed  up 
to  50  atmospheres,  but  no  very  accurate  measurements 
have  been  made.* 

From  the  fact  that  the  osmotic  constant  is  identical 
with  the  gas  constant,  both  calculated  for  the  volume 
containing  i  mole,  it  is  evident  that  the  equation 
pV=RT  is  applicable  to  both  the  dissolved  and  gaseous 
conditions,  p  being  the  osmotic  pressure  in  the  one  case 
and  that  of  the  gas  in  the  other,  while  in  both  cases 
V  is  the  volume  occupied  by  i  mole  and  R  is  a  constant 
depending  in  value  only  upon  the  units  chosen.  This 
fact  may  also  be  expressed  in  other  ways.  For  example, 
we  may  say  that  the  pressure  necessary  to  just  prevent 
pure  solvent  flowing  through  a  semipermeable  film  to  dilute 
a  solution  is  the  same  as  would  have  to  be  exerted  upon  the 
amount  of  substance  contained  in  the  solution  if  it  were  in 
the  gaseous  state,  at  the  same  temperature  and  occupying 
the  same  volume,  to  just  prevent  it  expanding;  provided,  of 
course,  that  the  formula  weight  is  the  same  in  both  states. 
Or,  with  the  same  proviso  as  to  formula  weight,  as  van't 
Hoff  originally  announced  the  law,  the  osmotic  pressure  oj 
a  substance  in  solution  is  the  same  pressure  as  that  amount 
0}  substance  would  exert  were  it  in  the  gaseous  state  at  the 
same  temperature  and  occupying  the  same  volume.  Pos- 

*  See  Raoult,  Zeit.  f.  phys.  Chem.,  27,  737,  1895. 


SOLUTIONS.  63 

sibly  a  better  form  of  this  law,  and  certainly  one  that  is 
more  general,  is  as  follows:  The  osmotic  pressure  exerted 
by  I  mole  of  substance  in  solution  is  the  same  as  the  gase- 
ous pressure  exerted  by  I  mole,  provided  the  conditions 
of  temperature  and  volume  are  identical. 

Based  upon  experiments  with  comparatively  weak 
solutions,  then,  we  may  define  the  formula  weight  in 
the  state  of  solution.  And  naturally  the  possible  forms 
of  expression  are  similar  to  those  for  the  gaseous  state. 
The  formula  weight  in  the  dissolved  state  is  that  weight 
which  in  the  volume  o)  approximately  22.4  liters  of  solvent 
will  exert  the  osmotic  pressure  o[  i  atmosphere  at  o°,  or  a 
corresponding  pressure  at  another  temperature  or  volume; 
or  is  the  weight  which  occupies  such  a  volume  oj  solvent 

pV 
that  R  in  the  equation  R=^=r  will  have  approximately 

the  value  of  84,800  when  p  is  expressed  in  grams  per 
square  centimeters  and  V  in  cubic  centimeters. 

One  other  thing  has  been  observed  in  working  with 
these  comparatively  weak  solutions,  i.e.,  that  the  osmotic 
pressure  is  independent  of  the  nature  of  the  semipermea- 
ble  film.  We  may  say,  then,  from  this,  and  the  fact  that 
the  nature  of  the  solvent  has  no  influence  upon  the 
pressure,  provided  the  formula  weight  is  the  same  in  all 
solvents,  the  osmotic  pressure  exerted  by  I  mole  in  any 
solvent  is  a  constant  so  long  as  the  temperatures  and  the 
volumes  of  the  solvents  are  alike. 

All  our  conclusions  thus  far  have  been  drawn  from 
experiments  upon  comparatively  weak  solutions,  and, 
indeed,  up  to  a  very  recent  date  all  our  knowledge  oi 
osmotic  pressure  was  derived  from  such  experiments  as 
have  been  mentioned  (together  with  very  many  others), 
where  the  concentration  never  exceeded  a  certain  small 


64  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

value.  Naturally,  this  left  much  uncertainty  as  to  our 
quantitative  law  for  osmotic  pressure,  and  to  the  limits 
within  which  the  observed  values  of  formula  weights 
held.  One  point  in  particular  here  is  the  question  as 
to  which  volume,  that  of  solution  or  of  solvent,  is  to  be 
considered  as  V  in  pV=  RT.  For  experimentally  it  has 
been  impossible  to  tell  which  should  be  chosen  owing 
to  the  fact  that  the  two  volumes  would  be  practically 
identical  for  such  dilute  solutions  as  we  have  considered. 

All  doubt  on  these  and  many  other  points  connected 
with  osmotic  pressure,  however,  have  been  removed 
very  recently  by  the  brilliant  work  of  Morse  and  Frazer,* 
who  have  measured  directly,  and  with  very  great  ac- 
curacy, the  osmotic  pressures  of  sugar  solutions  up  to  a 
concentration  of  342  grams  of  sugar  in  i  liter  of  water. 
Their  results,  a  summary  of  which  is  given  in  the  follow- 
ing table,  show  that  everything  said  above  for  dilute  solu- 
tions holds  true  for  the  more  concentrated  solutions  of 
sugar,  if  the  volume  V  in  pV=RT  is  taken  as  that  of  the 
pure  solvent.^ 

This  interpretation  of  volume  simplifies  the  laws  of 
osmotic  pressure  very  considerably,  for,  according  to  it,  the 
osmotic  pressure  of  a  mole  of  substance  in  a  liter  of  any 
solvent  is  the  same,  independent  of  the  possible  expansion 
or  contraction  caused  by  the  solution;  while  if  the  volume 
of  the  final  solution  were  the  significant  conception,  the 
osmotic  pressure  would  only  be  the  same  when  the 
changes  in  volume  caused  by  solution  are  the  same  in 
extent  and  direction.  Whether  other  substances  will  lead 


*  Am.  Chem.  Jour.,  34,  i,  July  1905. 

"j"  In  all  of  the  above  laws  and  definitions,  then,  this  volume  is  to  be 
taken  when  the  solution  in  question  is  so  strong  that  its  total  volume  is 
appreciably  different  from  that  of  the  pure  solvent. 


SOLUTIONS. 


OSMOTIC  PRESSURES  AND  FORMULA  WEIGHTS. 
SUGAR  SOLUTIONS  AT  ABOUT  2O°. 


Weight- 

Volume- 

Pressure  at  Same 

molar. 
Moles  in  1000 
Gr.  H2O 
(WO. 

molar. 
Moles  per 
Liter. 

Temperature.          **     ^(22.4+0.0824/1 

Gaseous. 

Osmotic  (P). 

P 

0-05 

o  .  04948 

I.  21 

1.26 

327-5 

O.  IO 

0.09794 

2.40 

2-44 

336.9 

0.20 

0.19192 

4.82 

4-78 

345-2 

0.25 

0.23748 

6.06 

6.05 

342.9 

0.30 

0.28213 

7.22 

7-23 

342.0 

0.40 

0.36886 

9.68 

9.66 

343  -1 

0.50 

0.45228 

12.07 

'12.09 

341-7 

0.6O 

0.53252 

14.58 

14.38 

347-1 

0.70 

0.60981 

17.16 

I7-03 

344-8 

0.80 

0.68428 

19.17 

19.38 

338.5 

0.89101 

0.75000 

21.  48 

21.21 

346.5 

0.90 

0.75610 

21-73 

21.  8l 

340.0 

1.  00 

0.82534 

24.27 

24-49 

339-2 

Mean 


341-2 


to  a  similar  law,  and  whether  still  more  concentrated 
solutions  of  sugar  will  continue  to  follow  this  law  (contrary 
to  gases,  where  high  pressures  fail  to  give  a  constant  when 
multiplied  by  the  volume),  are  questions  for  the  future. 
At  any  rate,  the  authors  have  so  perfected  their  method 
that  after  a  short  time  we  should  have  a  very  complete 
quantitative  knowledge  of  osmotic  pressure,  and  be  able 
to  state  exactly  the  limits  within  which  the  laws  that  we 
know  will  hold.  Until  that  time,  then,  we  may  consider 
the  laws  above  as  binding,  for  they  can  be  indirectly  con- 
firmed by  other  methods. 

Since  our  definitions  show  the  osmotic  pressure  of  i 
mole  of  substance  in  a  certain  volume,  it  is  simply  a 
matter  of  calculation  to  find  the  formula  weight  from 
the  osmotic  pressure  observed  for  a  solution  containing 


66  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

a  known  amount  of  substance.     In  general,  we  have  the 
proportion 


T 

where  22.4  -  is  the  osmotic  pressure  in  atmospheres  of 

a  solution  of  M  grams  in  i  liter  of  solvent,*  and  W  is  the 
number  of  grams  in  i  liter  of  solvent  which  gives  an  osmotic 
pressure  of  P  atmospheres.  For  example,  at  o°  C.,  a  2% 
solution  of  sugar,  i.e.,  about  20  grams  to  i  liter  of  water, 
gives  an  osmotic  pressure  equal  to  101.6  cms.  of  Hg, 
hence  the  formula  weight  of  sugar  can  be  found  from 
the  proportion 


.e.,  =335. 

Just  as  we  found  a  definition  of  formula  weight  in  the 
gaseous  state  from  the  work  done  by  expansion  (pp.  29- 
30),  we  can  also  find  one  adapted  to  the  dissolved  state 
from  the  work  necessary  for  the  removal  of  solute,  against 
the  osmotic  pressure,  from  the  solution.  Imagine  a  cyl- 
inder provided  with  a  semipermeable  piston,  water  being 
above  it  and  a  solution  below.  If  work  is  done  upon 
the  piston,  i.e.  if  it  is  lowered  into  the  solution,  pure 
solvent  will  be  removed  from  below  it  to  the  mass  of 
solvent  above.  And  since  pV=RT,  the  work  necessary 
to  remove"  the  amount  of  solvent  which  has  previously 

*  Since  i  mole  in  22.4  liters  of  solvent  gives  an  osmotic  pressure  of 
i  atmosphere  at  o°,  and  p=—,i  mole  in  i  liter  of  solvent  would  give  a 

T 
pressure  of  22.4  atmospheres,  or  at  T°  22.4  —  —  atmospheres. 


SOLUTIONS.  67 

contained  i  mole  will  be  equal  to  RT.  In  other  words, 
the  formula  weight  in  solution  is  that  weight  which  can 
be  separated  from  the  solvent  by  the  work  RT,  or,  what  is 
the  same  thing,  is  the  weight  which  was  previously  dis- 
solved in  the  volume  of  solvent  requiring  RT  units  of  work 
to  remove  reversibly  from  the  solution.  Any  method,  then, 
by  which  solvent  can  be  removed  from  a  solution  will 
serve  as  a  method  of  denning  formula  weight  (as  freezing, 
boiling,  etc.). 

In  addition  to  the  method  of  measuring  osmotic  pres- 
sure which  was  described  above,  there  is  one  that  is  so 
simple  and  at  the  same  time  so  striking  that  a  short 
description  of  it  will  possibly  make  the  conception  more 
clear  in  the  reader's  mind.  If  in  a  moderately  strong 
solution  of  copper  sulphate  we  place  a  drop  of  a  strong 
solution  of  potassium  ferrocyanide,  it  is  immediately  sur- 
rounded by  a  semipermeable  film  of  copper  ferrocyanide. 
We  have,  then,  a  semipermeable  film  of  copper  ferrocy- 
anide surrounding  a  strong  solution  of  potassium  ferro- 
cyanide. Since  the  ferrocyanide  is  stronger  than  the 
copper  sulphate  (i.e.,  contains  a  greater  number  of  formula 
weights  per  liter),  water  will  flow  into  the  bubble  with  a 
greater  force  than  it  will  flow  outward,  and  this  can  be 
proven  by  the  swelling  of  the  bubble  and  the  formation  of 
dark  streaks  in  the  copper  sulphate  solution,  which  is  con- 
centrated by  the  removal  of  water.  If  the  copper  sulphate 
drop  is  placed  in  the  potassium  ferrocyanide,  the  oppo- 
site effect  is  observed,  i.e.,  water  flows  outward  from  the 
bubble,  and  this  decreases  in  size.  By  this  method,  then, 
it  is  always  possible  to  show  an  equal  number  of  formula 
weights  to  the  liter,  for,  under  such  conditions,  no  change 
in  the  size  of  the  bubble  can  result.  And,  naturally, 
the  formula  weights  need  not  be  of  the  same  substance, 


68  PHYSICAL  CHEMISTRY  FOR.  ELECTRICAL  ENGINEERS. 

i.e.,  other  things  may  be  added  to  the  weaker  solution 
until  no  change  in  the  size  of  the  bubble  is  observed. 
After  such  an  addition,  then,  we  can  conclude  that  the 
sum  of  the  original  number  of  moles  to  the  liter  of  the 
weaker  solution  and  those  added  is  equal  to  the  original 
number  of  moles  in  the  stronger  solution. 

Vapor  pressure. — It  has  been  known  for  many  years 
that  the  vapor  pressure  of  a  liquid  is  always  depressed 
when  substance  is  dissolved  in  it.  It  was  not  until 
1887,  however,  that  Raoult  applied  chemical  conceptions 
to  the  physical  facts,  and  obtained  general  results.  Pro- 
ceeding in  a  way  similar  to  that  used  in  finding  a  defini- 
tion of  formula  weight  based  on  osmotic  pressure,  and 
always  using  the  formula  weight  of  the  solvent  as  found 
in  the  gaseous  state,  Raoult  found  that  the  vapor  pressure 
of  a  solution  is  related  to  that  of  the  pure  solvent  as  the 
number  of  moles  of  solvent  is  to  the  total  number  of  moles 
in  the  system,  i.e.,  of  solvent  plus  solute.  Our  definition  of 
formula  weight  in  the  dissolved  state  by  aid  of  vapor 
pressure,  then,  is  not  quite  as  simple  as  that  based  upon 
osmotic  pressure,  for  it  necessitates  a  knowledge  of  the 
formula  weight  of  the  solvent  when  in  the  gaseous  state. 
The  formula  weight  in  the  dissolved  state  is  that  weight 
which  when  dissolved  in  pp  moles  of  any  solvent  depresses 
its  vapor  pressure  1%.  And,  so  far  as  we  know,  the 
formula  weight  by  this  definition  agrees  in  each  case 
with  that  found  from  osmotic  pressure. 

Expressed  in  the  form  of  an  equation,  the  above  rela- 
tion between  the  vapor  pressures  and  the  number  of 
moles  may  be  written 

f^    N 
p~N+n' 


SOLUTION.  69 

where  p'  is  the  vapor  pressure  of  the  solution,  p  that 

of  the  pure  solvent,  »(=•—)  is  the  number  of  moles  of 
\     ml 

I     W\ 
dissolved    non-volatile  substance,   and  Nl  =  77 )  is    the 

number  of  moles  of  solvent  calculated  from  the  gaseous 
formula  weight.  This  relation  can  also  be  written  in 
other  forms,  i.e.,  can  readily  be  transformed  into 

P-V  _     » 

p        N+n' 


V    ~N- 


This  latter  form  is  very  useful  for  the  determination 
of  the  formula  weight  of  a  dissolved  substance,  the  other 
forms  being  better  adapted  for  the  calculation  of  vapor 
pressure  from  known  concentrations.  Thus  experiment 
shows  that  a  solution  of  2.47  grams  of  ethyl  benzoate 
in  100  grams  of  benzene  has  a  vapor  pressure  of  742.6  mm. 
of  Hg,  while  pure  benzene  shows  751.86  mm.,  both  at  80° 
C.  Since  ^=2.47,  Af  =78  (i.e.,  gaseous  C6H6),  W=ioo, 

p—tf     n 
p>  =  742.6,  and  p=  751.86,  we  find  from        .    =j^  that 

^=154,     while    the    formula    in    the    gaseous    state, 
,  leads  to  the  value  150.* 


*  For  the  vapor  pressure  of  a  system  of  two  non-miscible  liquids, 
as  well  as  for  those  cases  where  both  solvent  and  solute  are  volatile, 
see  "Elements,"  pp.  119,  158-161.  For  the  theoretical  deduction  of 
these  empirical  relations  from  the  conception  of  osmotic  pressure,  see 
pp.  161-167. 


70  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

Boiling-point.  —  Since  the  vapor  pressure  of  a  solution 
is  lower  than  that  of  the  pure  solvent,  and  the  boiling- 
point  is  that  temperature  at  which  the  vapor  pressure 
becomes  equal  to  the  atmospheric  pressure,  the  boiling- 
point  of  a  solution  must  be  higher  than  that  of  the  pure 
solvent.  Just  as  general  relations  were  found  for  the 
vapor  pressure  and  osmotic  pressure,  so  they  have  been 
found  for  the  boiling-point.  In  few  words,  it  has  been 
found  that  I  mole  oj  a  non-volatile  substance  dissolved 
in  100  grams  oj  a  solvent  gives  a  definite,  constant  increase 
oj  the  boiling-point,  which  depends  in  value  only  upon 
the  nature  oj  the  solvent.  This  so-called  molecular  in- 
crease oj  the  boiling-point  is  usually  designated  by  the 
letter  K.  The  formula  weight  oj  any  substance  in  the 
dissolved  state,  then,  is  that  weight  which  in  100  grams 
of  the  solvent  will  increase  its  boiling-point  K°.  The 
value  of  K  for  a  solvent  must  consequently  be  known 
before  it  is  possible  to  define  a  formula  weight  in  that 
solvent. 

Although  this  value  K  can  be  found  by  direct  observa- 
tion when  the  formula  weight  is  known,  i.e.  by  finding 
the  effect  on  the  boiling-point  of  a  small  amount  in 
100  grams  of  solvent,  and  then  calculating  this  weight 
to  the  formula  weight,  it  can  also  be  found  by  calculation, 
i.e.,  by  using  the  conception  of  osmotic  pressure.  In  this 
case  we  separate,  as  vapor,  the  solvent  from  the  solution, 
and  the  amount  of  work  necessary  for  this  must  be  equal 
to  that  necessary  for  an  osmotic  separation  (p.  66).  Since 
i  mole  in  a  liter  of  water  would  give  an  osmotic  pressure 


of  22.4X^-^  atmospheres  at  100°,  the  removal  of  the 
amount  of  solvent  containing  i  mole  would  necessitate 


SOLUTIONS.  71 

the    expenditure    of    22.4X-;       liter- atmospheres  *    as 

osmotic  work.  The  separation  of  this  weight  of  solvent 
as  vapor,  however,  since  the  latent  heat  of  evaporation  of 
i  gram  of  water  is  535.1  cals.,  would  require  thermal  work 
equal  to  1000X535.1  cals.  By  the  second  law  of  thermo- 
dynamics (p.  48,  third  statement),  then,  we  have  the 
relation. 

Work  done  _  Increase  of  temperature 

Heat  during  it,  in  terms  of  work     High  abs.  temperature ' 


22.4xfglit.at 


(1000X535.1)  0.041  lit.  at.     373+ K'1 

from  which  K=ioK'=$°.2,  for  K  refers  to  100  grams 
of  solvent.f  Some  other  values  o'f  K,  as  found  by  experi- 
ment (they  agree  with  the  calculated  ones),  are  as  fol- 
lows: Benzene,  26.70;  chloroform,  36.60;  carbon  disul- 
phide,  23.70;  ether,  21.5,  etc. 

Since  the  increase  of  the  boiling-point  is  known  for  a 
solution  containing  i  mole  in  100  grams,  the  molecular 
weight  of  a  substance  in  solution,  or  the  increase  of 
the  boiling-point  caused  by  the  solution  of  any  amount 
of  substance,  can  be  found  by  aid  of  a  simple  proportion. 


*  A  liter-atmosphere  is  the  work  done  when  the  pressure  of  the  atmos- 
phere on  i  square  decimeter  is  overcome  through  i  decimeter,  since  a 
liter  is  a  cubic  decimeter. 

t  A  very  general  relation  for  this  purpose  is  K=  —    — ,  where  T  is 

the  boiling-point  of  the  pure  solvent  and  w  is  its  latent  heat  of  evap- 
oration for  i  gram.     (See  "  Elements,"  pp. 

OF  THE 


72   PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 
We  have,  in  general, 

M\K\:W:M, 

where  M  is  the  formula  weight,  K  is  the  increase  when 
M  grams  are  present  in  100  grams  of  solvent,  and  Jt 
is  the  increase  for  W  grams  in  100  of  solvent.  Know- 
ing any  three  terms,  then,  the  other  one  may  be  readily 
calculated. 

Freezing-point. — The  fact  that  the  vapor  pressure  of 
a  solution  is  lower  than  that  of  the  pure  solvent  necessi- 
tates the  depression  of  the  freezing-point  of  a  solvent  in 
which  substance  is  dissolved,  i.e.,  causes  pure  solvent 


—t 


to  separate  from  the  solution  by  freezing  at  a  lower  tem- 
perature than  is  observed  for  the  pure  solvent.  This 
relationship,  perhaps,  is  not  so  obvious  as  is  the  one 
for  the  boiling-point,  but  a  glance  at  the  figure  above 
will  make  it  quite  clear.  Here  ww  is  the  vapor-pres- 
sure curve  for  water,  ss  that  for  the  solution,  and  I  that 
for  ice:  At  the  point  /=o°  C.,  ice  and  water  have  the 
same  vapor  pressure,  and  consequently  are  in  equilib- 
rium. The  solution  and  ice,  however,  will  only  be  in 


SOLUTIONS.  73 

equilibrium  at  the  temperature  corresponding  to  the 
point  of  intersection  of  their  curves,  so  that  the  freezing- 
point  of  the  solution  must  always  lie  below  that  of  the 
pure  solvent,  if  its  vapor  pressure  does.  And  the  more 
substance  there  is  in  solution  the  lower  will  be  the 
curve  ss,  and  the  lower  the  freezing-point,  i.e.,  the  point 
of  intersection. 

Exactly  as  with  the  boiling-point,  it  has  been  found 
that  I  mole  of  substance  dissolved  in  100  grams  0}  any 
solvent  will  depress  the  freezing-point  of  this  k°,  where 
the  value  of  k  depends  only  upon  the  nature  of  the  solvent. 
And  just  as  the  boiling-point  law  holds  only  when  pure 
solvent  and  no  solute  separates,  i.e.,  where  the  solute  is  non- 
volatile, so  here  this  law  only  holds  when  it  is  pure  solv- 
ent which  separates  in  the  solid  state*  One  thing  is 
to  be  observed  especially  in  regard  to  freezing.  If  the 
liquid  is  overcooled  and  solid  is  caused  to  separate  by 
stirring,  it  is  to  be  remembered  that  the  freezing-point 
observed  is  not  that  of  the  original  solution,  but  of  the 
stronger  solution  which  is  produced  by  the  loss  of  the 
solvent  solidifying,  for  the  freezing-point  of  a  solution 
is  that  temperature  at  which  it  exists  in  equilibrium  with 
the  solid  solvent,  f 

The  value  of  k  can  be  determined  here  in  a  similar  way 
to  that  used  for  K  in  the  boiling-point,  and  can  also  be  cal- 
culated by  an  analogous  method  of  reasoning.  Since  i 
mole  in  a  liter  of  water  at  o°  exerts  an  osmotic  pressure  of 
22.4  atmospheres,  the  osmotic  work  necessary  to  remove 
the  solvent  which  has  previously  contained  i  mole  will  be 


*  In  case  these  conditions  are  not  fulfilled,  it  is  still  possible  to  get 
results,  but  the  calculations  are  necessarily  more  complicated. 

f  For  the  correction  to  be  made  here,  see  "  Elements,"  pp.  185-186. 


74  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

22.4  liter-  atmospheres.  The  heat  involved  in  the  freezing- 
out  of  i  liter  of  water,  however,  would  be  1000X80  cals., 
where  80  cals.  is  the  latent  heat  of  solidification  of  i  gram 
of  water.  By  the  second  law  of  thermodynamics  (p.  48, 
second  statement),  then,  we  have 

Work  done  Lowering  of  temperature 

Heat  during  it,  in  terms  of  work      High  abs.  temperature 

22.4  lit.  at.  k' 

*  * 


80000X0.041  lit.  at.  "273' 


from    which    &=io&'  =  i8.6.*    Other    values    of    k   are 
as  follows:    Acetic  acid,  38.8;    benzene,  49.0;    phenol, 
75.0,  etc. 
Here,  also,  we  have  the  general  relation 


where  M  grams  in  TOO  grams  of  solvent  produce  a  depres- 
sion of  k°  in  the  freezing-point  and  W  grams  in  the  same 
weight  leads  to  a  depression  of  At°.  And,  just  as  above, 
when  any  three  of  these  are  known  the  fourth  may  be 
calculated. 

Coefficient  of  distribution.  —  One  exceedingly  impor- 
tant relation  has  been  observed  as  to  the  distribution 
of  a  substance  between  two  non-miscible  solvents.  This 
relation,  indeed,  is  the  one  mentioned  above  (p.  61),  by 
which  the  change  in  the  formula  weight  during  transi- 
tion from  one  solvent  to  another  is  detected.  When  a 


*  A  general  relation,  corresponding  to  the  one  for  the  boiling-point,  is 

&=  — ,  where  w,  here,  is  the  heat  of  solidification  of  i  gram  of 

solvent  and  T  is  its  freezing-point.     (See  "  Elements,"  pp.  177-179.) 


SOLUTIONS.  75 

solution  is  agitated  with  an  equal  volume  of  another 
solvent,  which  can  dissolve  the  solute,  but  does  not 
form  a  homogeneous  mixture  with  the  first  solvent,  it  is 
found  that  the  distribution  in  the  two  layers  is  either 
the  same  for  all  original  concentrations  of  solute  in  the 
first  solvent,  or  differs  with  that  concentration.  Thus 
when,  by  our  definitions,  the  formula  weight  is  the 
same  in  the  two  solvents  (succinic  acid  dissolved  in 
water  and  shaken  with  ether,  for  example)  the  ratio 
of  the  concentrations  hi  the  two  layers  is  independent 
of  the  original  amount  dissolved  in  the  first  solvent. 
When  the  formula  weights,  by  definition,  differ  in  the 
two  solvents,  however,  the  ratio  depends  upon  the  original 
concentration,  and  in  such  a  way,  as  we  shall  see  later, 
that  it  is  possible  for  us  to  calculate  the  relation  of  the 
formula  weight  in  one  to  that  in  the  other. 

This  relation  is  not  restricted  to  a  solution  containing 
one  solute,  for  it  has  been  observed  that  when  several  solutes 
are  present  each  is  distributed  as  if  it  alone  were  present, 
and  its  behavior  is  entirely  unaffected  by  the  others. 


Since  all  the  above-mentioned  properties  of  a  solution, 
i.e.  the  freezing-point,  boiling-point,  vapor  pressure, 
and  osmotic  pressure,  depend  upon  the  concentration  of 
substance  dissolved,  and  since  by  definition  the  effect  of 
i  mole  in  the  dissolved  state  is  known,  it  is  possible  to  go 
from  a  result  by  one  method  to  the  results  by  the  others 
when  we  know  the  concentration  of  the  solution  in  moles 
per  liter.  Thus,  suppose  a  water  solution  gives  an  osmotic 
pressure  of  i  atmosphere  at  o°,  and,  assuming  the  formula 
weight  to  be  independent  of  temperature,  we  wish  to 
find  the  freezing-point,  boiling-point,  and  vapor  pressure 
of  the  solution.  Since  i  mole  dissolved  in  i  liter  of  water 


76  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

or  any  other  solvent  at  o°  gives  an  osmotic  pressure  of 
22.4  atmospheres,  in  a  solution  with  an  osmotic  pressure 
of  i  atmosphere  we  must  have  x  moles  per  liter,  the 
value  of  x  being  determinable  from  the  proportion 

22.4:1  ::i:x. 

The  boiling-point  of  this  solution,  then,  if  the  solvent  be 
water,  will  be  ( —  X5°.2J+ioo°,  and  the  freezing-point 

—  ( —  Xi8°.6j;    while  the  vapor  pressure  will  be  p'  in 
1000 

i f  o 

T  =  ~~    >  where  p  is  the  vapor  pressure   of   pure 

P     1000 

*  I         /yt 

18 
water  at  o°. 

Or,  if  the  vapor  pressures  of  solvent  and  solution  are 
given,  we  can  first  find  the  ratio  of  moles  of  solute 
to  moles  of  solvent  (based  on  the  gaseous  formula 

weight)  from  ,  =-r-=,-  and  then  calculate,  for  example, 
the  number  of  moles  dissolved  in  i  liter  of  water 

/•  AT         I0°° 

(i.e.,  where  N= — ^-  =  55 

\  I O 

Electrolytic  dissociation  or  ionization. — In  obtaining 
our  definitions  of  formula  weight,  as  given  above,  we 
have  assumed  in  each  case,  as  a  temparory  supposition, 
that  the  formula  weight  in  the  dissolved  state  is  identical 
with  that  in  the  gaseous  state.  In  this  way  a  constant 
relation  has  been  observed  for  a  large  number  of  substances 
and  our  temporary  assumption  is  justified.  Sooner  or  later, 
in  all  cases,  however,  apparently  abnormal  results  (the 


SOLUTIONS.  77 

formula  weight  being  greater  than  that  in  the  gaseous  state) 
are  observed,  and  it  becomes  necessary  to  decide  whether 
the  definitions  so  obtained  are  incorrect;  or  whether  in 
these  abnormal  cases  there  is  not  some  specific  influence 
which  has  not  been  considered.  Since  as  a  rule  these 
abnormalities  disappear  when  the  substance  is  dissolved 
in  another  solvent,  the  relation  then  being  the  same  as 
for  substances  which  behave  normally  in  all  solvents,  the 
only  conclusion  possible  is  that  the  definitions  for  formula 
weight  in  all  these  cases  are  correct,  and  that  the  apparently 
abnormal  results  are  really  due  to  a  change  in  the  formula 
weight.  This  conclusion,  indeed,  is  the  only  one  possible 
in  such  cases,  for  all  definitions  of  formula  weight  in 
solution,  independent  of  the  principle  upon  which  they 
are  based,  give  sensibly  the  same  formula  weight  when 
carried  out  with  the  same  degree  of  accuracy. 

In  other  cases,  and  these  form  a  very  large  class  composed 
of  inorganic  salts,  bases,  and  acids,  the  same  question  as 
was  considered  above  also  arises,  but  here  the  formula 
weight  (by  definition)  appears  smaller  than  the  gaseous 
formula  weight.  Again  the  abnormality  may  be  due  to 
incorrect  definition,  or  to  the  fact  that  a  specific  action 
causes  the  formula  weight  to  change.  And  again  here 
we  find  that  the  solvent  has  a  great  influence,  i.e.,  in  some 
solvents  abnormal  results  are  observed,  while  in  others 
the  behavior  is  quite  normal  (according  to  definition). 
Thus  Arrhenius  observed  that  all  those  substances,  and 
only  those,  which  give  abnormally  large  osmotic  pressures 
in  solution  are  capable  of  conducting  the  electric  current, 
and  ij  they  are  dissolved  in  other  solvents,  in  which  they 
behave  normally,  they  lose  this  power. 

Arrhenius  determined  the  electrical  conductivity  of 
such  solutions  in  terms  of  molecular  conductivity;  the 


?8  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

molecular  conductivity  of  a  solution  being  denned  as 
the  reciprocal  of  the  resistance  (in  ohms)  of  the  volume 
of  liquid  which  contains  one  formula  weight  of  the  sub- 
stance, i.e.,  the  weight  of  the  generally  accepted  formula, 
the  electrodes  being  i  cm.  apart  and  large  enough  to 
contain  between  them  the  entire  amount  of  solution. 
This  value,  naturally,  is  not  found  directly,  but  is  calcu- 
lated from  that  value  found  for  a  centimeter  cube  of  the 
solution.  (See  Chapter  VII.)  In  this  way,  always  having 
i  mole  (according  to  the  accepted  formula)  between  the 
electrodes,  he  found  that  the  more  dilute  the  solution  the 
greater  is  the  molecular  conductivity.  In  many  cases, 
indeed,  he  was  able  to  reach  such  a  dilution  that  the  molec- 
ular conductivity  attained  a  maximum  value,  which  is 
unaffected  by  further  dilution.  This  molecular  conduc- 
tivity at  infinite  dilution,  as  it  is  called,  is  designated  by 
the  term  /z^,  that  value  for  any  dilution  V  being  desig- 
nated by  fjLv 

From  this  it  is  apparent  that  the  solution  undergoes 
some  kind  of  a  change  as  the  result  of  dilution;  and  the 
investigation  of  such  solutions  at  various  dilutions  shows, 
indeed,  that  the  formula  weight  (according  to  definition) 
also  changes  with  the  dilution,  the  formula  weight  de- 
creasing to  a  minimum,  constant  value,  which  for  binary 
electrolytes  is  one-half  the  formula  weight  of  the  substance 
dissolved.  We  may  conclude,  then,  that  the  breaking 
down  of  the  formula  weight  of  a  substance  in  a  solution 
is  very  intimately  connected  with  the  power  it  possesses 
of  conducting  the  electric  current. 

These  facts  formed  the  starting-point  of  what  is  known 
at  present  as  the  "theory  of  electrolytic  dissociation." 

As  this  theory  to-day  is  much  misunderstood  by  many, 
and  is  the  subject  of  much  speculation  on  the  part  of 


SOLUTIONS.  79 

others,  it  will  be  necessary  for  us  to  consider  carefully 
just  what  is  fact  and  what  assumption,  and  to  see  clearly 
which  portions  are  hypothetical  and  which  are  destined 
to  remain  under  any  hypothesis  or  lack  of  hypothesis; 
in  other  words,  which  are  experimental  facts.  It  may  be 
said,  however,  that  that  which  is  hypothesis  in  this  theory 
is  unessential,  as  far  as  the  use  of  the  data  is  concerned, 
and  the  only  hypothesis  present,  as  we  shall  consider  it, 
is  that  inherent  in  the  terminology,  which  is  a  relic  of  the 
atomistic  hypothesis  and  utterly  beyond  our  power  either 
to  prove  or  disprove. 

The  salient  facts  which  have  been  grouped  in  this 
theory,  for  it  is  a  theory  in  the  sense  that  it  is  a  law  of 
nature  holding  between  certain  limits,  although  these  are 
not  as  yet  definitely  fixed,  are  as  follows: 

(1)  The  molecular  conductivity  of  certain  substances 
in  water  is  found  to  increase  up  to  a  maximum,  constant 
value,  and  this  increase  is  the  result  of  dilution. 

(2)  Those  solutions  which  conduct  the  current  also 
give  abnormal  osmotic  pressures,  freezing-points,  boiling- 
points,  and  vapor  pressures ;   in  other  words,  the  formula 
weight  (according  to  the  above  definitions)  decreases  with 
increased  dilution,  and  finally  reaches  a  minimum  value, 
which,  for  binary  electrolytes,  is  one-half  the  accepted 
formula  weight  of  the  substance. 

(3)  Those    substances   which    in    water   conduct   the 
current  and  give  abnormal  osmotic  pressures,  depressions 
of  the  freezing-point  and  vapor  pressure,  and  increases 
of  the  boiling-point,  give  normal  values  when  dissolved 
in  other  solvents  in  which  they  do  not  conduct. 

(4)  The  nearer  the  value  of  fiv  is  to  that  of  u^  the 
more   abnormal  the  value  of  the  osmotic  pressure,  etc. 
(formula  weight);  of  the  solution.    And  the  solution  for 


8o  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

which  fjLw  is  found  also  gives  the  maximum  osmotic 
pressure,  i.e.,  the  minimum  formula  weight. 

(5)  The  molecular  conductivity  of  a  solution  at  infinite 
dilution  is  an  additive  value,  i.e.,  is  equal  to  the  sum  of 
the  conductivities  of  the  substances  of  which  it  is  com- 
posed.    The  meaning  of  this  is  as  follows:   The  molecu- 
lar   conductivity    at    infinite    dilution    of,    for    example, 
potassium  chloride  plus  that  of  nitric  acid  minus  that  of 
potassium  nitrate  is  found  to  be  equal  to  that  of  hydro- 
chloric acid.     In  other  words, 

^ooKCl  +  /*ooHN03  -  /"ooKN03  =  /*ooHCl  • 

For  this  to  be  true,  and  it  is  true  in  general  for  all  sub- 
stances, it  is  necessary  that  the  molecular  conductivity 
of  such  a  substance  in  solution  be  the  sum  of  two  values 
which  are  independent  each  of  the  other.  Chlorine, 
for  example,  as  the  constituent  of  an  electrolyte,  at  the 
dilution  giving  p^,  has  the  same  conducting  effect  when 
part  of  a  compound  with  one  element  as  it  has  when 
combined  with  any  other.  It  is  possible,  ihen,  to  find 
the  value  of  p^  for  any  binary  electrolyte  when  the 
values  for  the  elements  composing  it  are  known.  In 
other  words,  the  conductivities  of  the  solution  as  pro- 
duced by  the  presence  of  any  element  can  be  calculated; 
and  from  these  values,  by  summation,  the  value  of  /^ 
for  any  binary  electrolyte  can  be  found. 

(6)  When  a  solution  is  electrolyzed,  the  products  of 
electrolysis  appear  instantaneously  at  the  electrodes  so 
soon  as  the  circuit  is  completed.     This  indicates  (since 
the  solvent,  water,  does  not  conduct  beyond  a  very  small 
extent)    that  whatever  does  carry  the  current  through 
the  liquid   is   charged   with   electricity   even   before  the 
current  is  applied,  for  the  conduction  is  due  to  the  dis- 


SOLUTIONS.  8 1 

solved  substance,  and  the  speed  of  movement  of  the  sub- 
stance can  be  measured,  so  that  it  is  no  question  of  matter 
being  electrically  charged  at  one  electrode  before  carry- 
ing this  charge  bodily  through  the  solution  to  the  other. 
(See  Chapter  VII.)  Further,  it  is  observed  that  the  same 
amount  of  electricity,  96,540  coulombs,  is  necessary  for 
the  separation  of  one  equivalent  weight  (in  grams)  of 
any  element;  in  other  words,  that  96,540  coulombs  of 
electricity  are  transported  through  the  liquid  with  each 
equivalent  weight  (in  grams)  of  an  element.  (Faraday's 
Law,  see  Chapter  VII.) 

(7)  The  properties  of  electrolytes  are  found  to  be  the 
sum  of  the  properties  of  the  products  observed  during 
electrolysis.  Thus  any  solution  giving  off  chlorine  on 
electrolysis,  excluding  secondary  reactions,  will  precipi- 
tate silver  from  its  solution  as  the  chloride.  And  if 
chlorine  cannot  be  produced  in  any  way  by  the  elec- 
trolysis, silver  will  not  be  precipitated  as  chloride  from 
its  solutions.  And,  on  the  other  hand,  silver  is  only 
precipitated  by  chlorine  when  contained  in  a  solution 
from  which  silver  can  be  deposited  by  the  current  by 
primary  action. 

The  catalytic  effect  of  acids  on  the  inversion  of  sugar 
as  well  as  on  the  decomposition  of  methyl  acetate  is 

found  to  be  proportional  to  the  ratio  —  for  the   acid; 

/*oo 

and  when  a  large  amount  of  a  salt  of  this  acid  is  added 
to  the  acid  this  effect  is  decreased.     But  this  is  only 
true  when  the  salt  added  is  an  electrolyte. 
All  copper  solutions,  when  very  dilute,  show  the  same 

blue  color,  and  this  also  depends  upon  the  ratio  — , 

"oo 

and   can  also  be  changed,  as  the    effect    of  acids  was 


82  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

above,  by  the  addition  of  a  large  amount  of  an  elec- 
trolyte which  contains  the  same  acid  radical  as  the  cop- 
per salt  in  question. 

Further,  when  the  colored  copper  solution  is  super- 
imposed upon  a  colorless  solution  of  another  salt,  the 
blue  color  boundary  is  observed  to  move  with  the  current, 
i.e.,  to  the  cathode,  where  copper  is  deposited.  Hence 
the  substance  which  is  moved  in  this  direction  contains 
only  copper,  the  negative  radical  separating  at  the  anode. 
In  other  words,  copper  in  solution,  when  it  conducts  the 
current,  is  blue. 

8.  Observation  shows  that  when  an  element  is  sepa- 
rated on  one  electrode,  anode  or  cathode,  it  is  always 
separated  on  that  one  by  primary  action ;  in  other  words, 
the  sign  of  the  electricity  transported  by  an  element  is 
always  the  same.  And  unless  an  element  in  the  pure 
state,  when  dissolved  in  water,  reacts  with  the  water  it  does 
not  conduct  the  current.  This  circumstance  is  assumed 
to  be  due  to  the  fact  that  only  one  kind  of  electricity 
could  be  carried  by  the  substance,  and  hence  it  pro- 
duces no  conduction. 


The  question  now  arises  as  to  what  theory  can  be 
found  to  correlate  these  facts  and  observations  so  that 
the  generalization  thus  obtained  may  be  employed  to 
foresee  other  facts,  and  applied  to  other  observations, 
that  they,  in  their  turn,  may  be  elucidated  and  general- 
ized. By  the  word  theory,  then,  we  do  not  mean  a 
hypothesis,  in  which  something  not  observed  is  added  to 
the  facts  to  "explain  "  them,  but  only  a  generalization  of 
observed  facts.  In  other  words,  what  law  of  nature,  hold- 
ing within  definite,  if  small,  limits,  can  be  obtained  from 
the  above  experimental  facts  when  considered  together? 


SOLUTIONS.  83 

The  generalization  which  has  been  made  from  these 
facts  is  known  as  the  theory  of  electrolytic  dissociation, 
and,  considering  those  portions  which  are  free  from 
hypothesis  and  fulfil  the  above  conditions,  in  other  words, 
omitting  the  hypothetical  portions  which  it  has  attained 
since  the  time  of  its  inception,  we  find  in  it,  within  cer- 
tain limits,  a  definite  law  of  nature. 

The  principal  points  of  this  theory  are  summarized 
below  in  brief  form,  and  will  each  be  expanded  in  the 
later  portions  of  the  book. 

A  substance  in  solution,  which  conducts  the  electric 
current,  is  dissociated  or  ionized  into  its  constituents, 
and  these  constituents,  when  secondary  actions  are  ex- 
cluded, appear  at  the  electrodes  during  electrolysis. 
The  extent  of  the  ionization  or  dissociation  in  any 
solution  being  given  at  the  dilution  V  (number  of  liters 
in  which  i  mole,  according  to  the  accepted  formula,  is 

dissolved)  by  the  ratio  —  =  a. 

/JflO 

These  products  of  ionization  or  dissociation  are  charged 
with  electricity,  96,540  coulombs  being  carried  by  the 
gram  equivalent  of  any  element  (see  (6)  above).  A 
further  proof  of  this  charged  state  of  ionized  matter  is 
given  by  the  fact  that  not  only  is  the  current  carried  by 
a  solution  dependent  upon  the  number  of  gram  equiva- 
lents transported,  but,  as  we  shall  see  later,  any  other 
means  of  depositing  the  constituents  of  the  solution 
upon  the  electrodes  liberates  an  amount  of  electricity 
which  depends  also  upon  the  number  of  gram  equiva- 
lents deposited.  And  all  cells  in  order  to  give  a  current 
must  contain  electrolytes,  i.e.,  solutions  which  are  ionized. 

Since  a  solution  which  by  conductivity  is  shown  to  be 
completely  ionized,  or  practically  so,  leads  to  a  formula 


84  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

weight,  by  osmotic  pressure  or  any  of  the  other  methods, 
of  one-half  the  value  expressed  by  the  formula  weight, 
then,  from  the  case  of  hydrochloric  acid  in  solution,  where 
we  can  designate  the  process  by  the  equation 


the  formula  weight  of  the  hydrogen  and  chlorine  in  the 
ionic  state,  according  to  our  definition  of  formula  weight, 
must  be  synonymous  with  the  combining  weight. 

The  ionic  state,  then,  is  an  allotropic  form  of  the  or- 
dinary state  of  the  constituents,  and  differs  from  that  in 
being  charged  with  electricity,  in  having  less  energy  than 
when  in  the  gaseous  state,  and  in  always  being  trans- 
formed into  the  ordinary  state  on  the  loss  of  its  charge 
of  electricity. 

Since  the  constituents  in  the  case  already  mentioned, 
and  in  general  in  all  cases,  show  a  formula  weight  (by 
the  definitions)  which  is  the  same  as  the  combining  weight, 
it  is  possible  to  determine  a,  the  degree  of  ionization,  by 
osmotic-pressure,  etc.,  measurements,  or  from  the  average 
formula  weight  of  the  substance  in  solution,  as  determined 
by  osmotic  pressure  or  any  of  the  other  methods.  If, 
for  example,  we  start  with  one  formula  weight  of  hydro- 
chloric acid  in  a  solution,  and  a  moles  of  it  are  ionized, 
the  total  number  of  moles  will  consist  of  (i  —  a)  of  un- 
ionized HC1  and  a  moles  each  of  H'  and  Cl'  (where 
the  dot  indicates  positive  electricity  as  the  charge  and 
the  accent  negative).  The  total  number  of  moles  in  the 
volume  of  the  solution  will  go  then  from  i  to  (i  —a)  +  2a, 
i.e.  i  +a,  and  the  ratio  of  osmotic  pressure  when  entirely 
un-ionized  to  that  when  partially  ionized  will  be  the  same 
as  this.  In  other  words,  if  the  formula  weight  in  a  certain 
volume  should  give  the  osmotic  pressure  p',  it  will  give, 


SOLUTIONS.  85 

when  ionized  to  the  extent  a,  the  pressure  p=(i+a)p'. 
Since  the  number  of  moles  (by  definition)  shown  by  the 
same  weight  is  thus  increased,  the  formula  weight  will  be 
smaller,  and  the  relation  between  the  two  values  of  the  for- 
mula weight  will  be  M'(L  +a)=M,  where  the  M  refers  to 
substance  if  it  were  un-ionized,  i.e.,  is  the  accepted  formula 
weight  of  the  substance,  and  Mf  is  the  formula  weight  (by 
definition)  observed  in  the  dissolved  state. 

Just  as  with  gaseous  dissociation,  the  ionization  of  a 
substance  in  solution  is  affected  by  the  presence  of  one 
of  the  products  of  the  ionization,  and  later,  when  we 
consider  the  quantitative  effects  for  gases,  we  shall  study 
the  quantitative  effect  for  substances  in  solution. 

Owing  to  the  fact  that  the  constituents  produced  by 
the  ionization  of  a  substance  in  solution  are  called  ions 
(in  the  Faraday  sense  of  charged  atoms)  it  is  usually 
assumed  that  ionized  matter  also  has  an  atomic  structure. 
As  this  is  hypothesis,  if  we  are  to  follow  our  plan,  we 
must  either  use  the  word  with  an  altered  meaning  or 
employ  another  word  representing  the  same  facts  in  its 
place.  We  shall  use  the  word  ionization  here  only  in  the 

sense  that  it  is  expressive  of  the  experimental  relation  — -, 

and  employ  the  expression  ionized  matter  to  designate 
all  that  is  ever  legitimately  included  in  the  word  ion,  i.e., 
all  the  facts  and  none  of  the  hypotheses. 

Summarizing  our  argument,  the  application  of  the 
experimental  definitions  of  formula  weight  in  solution 
(derived  as  given  above)  indicates  that  certain  substances 
are  decomposed  in  certain  solvents,  the  fraction  decom- 
posed being  a  in  the  expressions  M'(i+a)  =  M  (accord- 

u 

ing  to  any  of  the  methods)  and  a  =  — ,  and  increasing 


S6  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

with  the  dilution  up  to  the  value  which  gives  M=2M'. 
This  is,  of  course,  only  true  for  substances  giving  at 
the  maximum  dilution  a  formula  weight  of  one-half  the 
generally  accepted  one;  in  general  the  fraction  decom- 
posed can  be  found  from  [(i  —  a)+na]M'  =  M)  where 

M 
at  the  maximum  dilution  n=—r     That  this  is  really 

the  result  of  a  decomposition,  and  not  merely  the  failure 
in  these  cases  of  the  definitions  of  formula  weight  in 
solution,  is  evidenced  by  the  above  facts  and  many 
others,  given  later.  And  that  this  ionized  matter  which 
is  formed  is  electrically  charged  is  also  not  to  be  doubted, 
as  well  from  the  above  facts  as  from  the  general  agree- 
ment of  the  results  by  electrical  and  other  methods. 

It  is  always  to  be  remembered,  then,  that  when  we  speak 
of  ionization  we  mean  something  which  can  be  defined 
in  terms  of  experiment,  and  is  free  from  hypothesis. 
And  the  same  is  true  of  ionized  matter,  so  long  as  we 
do  not  assume  for  it  a  certain  structure  such  as  is  naturally 
assumed  in  the  impression  made  by  the  expressions  "  an 
ion  "  or  "  the  ions." 

Later  we  shall  find  that  starting  with  this  conception 
for  a  simple  substance  we  can  derive  other  experimental 
definitions,  not  only  for  ionization,  but  also  for  the 
amount  of  any  one  definite  kind  of  ionized  matter  which 
is  present  with  any  number  of  other  kinds. 

One  fact  may  be  mentioned  here  which  indicates  what 
a  very  marked  difference  dilution  makes  in  the  behavior 
of  a  substance,  and  which  decidedly  supports  the  con- 
clusions we  have  just  drawn.  Although  hydrochloric 
acid  is  more  volatile  than  hydrocyanic  acid,  it  has  been 
observed  that  from  a  mixture  of  the  dilute  acids  (o.i 
molar  of  HC1)  it  is  possible  to  distil  the  HCN  quantita- 


SOLUTIONS.  87 

lively  (provided  the  dilution  of  the  HC1  is  retained  at 
about  this  value  by  the  frequent  replacement  of  the 
water  lost).  In  the  light  of  the  above  theory  the  difference 

between  the  two  acids  in  solution  is  that  while  —  is 

000 

nearly  equal  to  i  for  HC1,  it  is  very  small  for  HCN.  In 
other  words,  HC1  is  composed  principally  of  the  ionized 
constituents  H*  and  Cl',  which  cannot  produce  HC1  gas 
without  going  through  the  state  HC1  in  solution,  and  that 
is  prevented  by  the  nearly  constant  dilution  which  is 
retained  during  the  distillation.  Any  gaseous  substance, 
then,  which  in  solution  is  largely  ionized  is  more  difficult 
to  distil  from  the  liquid  than  an  un-ionized  or  less  ionized 
one.  The  HCN,  being  dissolved  and  retained  in  this 
state  in  solution,  can  be  expelled  readily  just  as  any 
other  gas  which  undergoes  no  great  change  in  solution. 
This  method,  indeed,  was  discovered  as  the  result  of  such 
theoretical  reasoning,  and  it  is  but  one  example  of  the 
many  practical  applications  of  the  above  generalization.* 
It  is  not  to  be  imagined  that  the  facts  mentioned  above 
are  the  only  ones  leading  to  these  conclusions,  for  later, 
throughout  our  work,  we  shall  find  occasion  to  consider 
other  things  which  will  confirm  each  of  the  steps  leading 
to  the  final  conclusion.  In  other  words,  it  is  not  to  be 
thought  that  the  whole  theory  has  been  described  in  this 
place,  or  that,  because  some  of  the  points  mentioned  are 
not  clear,  the  theory  itself  is  to  be  condemned,  for  many 
of  the  points  can  only  be  brought  out  after  considering 
certain  other  methods  which  will  enlarge  our  horizon. 
It  may  be  said,  however,  that  these  further  aids  but 

*  For  details  of  the  separation  see  Richards  and  Singer,  Am.  Chem. 
J.,  27,  205,  1902. 


88  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

confirm  and  make  more  evident  the  truth  of  the  con- 
clusions we  have  arrived  at.  At  the  same  time  we  must 
not  forget  that  we  have  been  speaking  of  this  subject  as 
lying  within  certain  limits,  and  so  cannot  expect  our  con- 
clusions to  hold  outside  of  them,  nor  to  condemn  them 
because  they  do  not.  The  relation  of  substances  in  non- 
aqueous  solvents  to  a  certain  extent  is  different,  and  con 
sequently  these  conclusions  could  not  be  expected  to  hold. 
As  a  matter  of  fact,  the  conduction  relations  for  these 
solutions  are  so  utterly  different  from  the  aqueous  ones 
that  it  would  be  impossible  to  attempt  to  consider  them 
together  in  the  light  of  our  present  knowledge.  All  of  these 
points  will  be  discussed  more  fully  later,  however,  and 
the  limits  stated,  within  which  our  conclusions  in  general 
will  hold.  It  is  to  be  remembered,  though,  that  simply 
because  our  theory  does  not  hold  for  solutions  in  certain 
non-aqueous  solvents  (solutions  which  show  no  similarity 
in  behavior  to  the  aqueous  ones,  and  which  may  or  may 
not  be  solutions  as  we  consider  them,  but  may  involve 
an  entire  rearrangement  of  the  composition  of  the  solvent, 
or  solute,  or  both),  it  should  not  be  considered  as  false  and 
of  little  use,  for  the  two  kinds  of  systems  are  so  different 
that  it  would  be  impossible  to  imagine  from  our  pres- 
ent knowledge  that  both  are  subject  to  the  same  laws. 

The  values  for  a,  the  degree  of  ionization,  for  a  few 
electrolytes  are  given  below  for  varying  conditions.  These 
are  the  values  as  found  from  the  ratio  of  molecular  con- 
ductivities, since  that  method  is  apparently  the  most 
delicate  one  which  we  possess  for  this  purpose. 

Naturally,  instead  of  first  finding  the  formula  weight 
in  solution,  by  aid  of  one  of  the  practical  definitions, 
and  then  calculating  a  from  the  relation  of  this  value 
to  the  generally  accepted  formula  weight,  we  can  find 


SOLUTIONS. 


89 


4 

S 

16 


16 
64 


16 

64 

512 

HBr 

V25° 

a 

0.897 
0.932 
0.950 
0.965 


DEGREES  OF  IONIZATION. 

AgN03 

AgNO3 

25° 

V 

60° 

a 

a 

0.828 

16 

0.841 

0.899 

64 

0.909 

0.962 

512 

0.964 

40° 

HC1 

0.832 

a 

0.904 

2 

0.876 

0.965 

16 

o-955 

HI 

25° 

KC1 
V         25° 

NaCl 

25° 

NH.Q 

LiCl 

25° 

a 

a 

a 

a 

a 

0.895 

2          .... 

o-737 

0.926 

10       0.86 

0.842 

0.852 

0.803 

0-945 

100       0.94 

o-937 

0.94 

0.907 

0.963 

1000       0.98 
i  oooo       o  .  99  3 

0.982 

0.979 



16667       0.006 

directly  the  number  of  moles  present,  when  we  have 
started  with  i  formula  weight  (the  accepted  value)  in 
a  certain  volume  of  liquid.  If  the  commonly  accepted 
formula  weight  in  a  certain  volume  should  give  the  osmotic 
pressure  P,  and  the  substance  is  ionized,  the  observed 
osmotic  pressure  would  be  [(i  — a)+wa]P,  and  the  vapor 
pressure  would  be  that  calculated  for  w[(i  —a)  +  wa],  where 
n  is  the  number  of  moles  which  would  be  present  with- 
out ionization.  In  a  corresponding  way,  also,  the  boil- 
ing-point of  a  solution  containing  i  accepted  formula 
weight  in  100  grams  of  solvent  would  be  [(i  —  a)+na]K°, 
higher  than  the  pure  solvent,  and  the  freezing-point 
would  be  depressed  by  [(i  —  ct)+  na]k°.  The  freezing- 
point  depression  produced  by  the  dissociation  of  i  ac- 
cepted formula  weight  in  100  grams  of  solvent  is  usu- 
ally designated  as  the  molecular  depression  of  the  sub- 
stance. Thus  a  0.0107  m°lar  solution  of  KOH  (assum- 


90  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

ing  this  formula)  depresses  the  freezing-point  o°.0388. 
Since  I  mole  in  solution  depresses  the  freezing-point 
1 8°.  6  when  dissolved  in  100  grams  of  water,  the  molecu- 
lar depression  of  our  solution  is 

0.0388X10 

=  36.261; 

w.wiwy 

~lf\    r>f\t 

hence 

i.e., 

The  thermal  relations  of  electrolytes.  —  Two  salt 
solutions  which  are  so  dilute  that  the  ratio  —=i  (p.  83) 

do  not  evolve  or  absorb  heat  when  mixed,  provided  no 
chemical  reaction  takes  place  between  them. 

This  fact  was  first  observed  by  Hess  and  has  been  con- 
firmed by  all  observers  since. 

Another  experimental  fact  observed  to  hold  for  solu- 
tions of  electrolytes  is  as  follows:  When  an  acid  is  neu- 
tralized by  a  base,  both  being  in  so  great  a  dilution  that 

for  each  —  =  i,  which  is  also  true  for  the  salt  formed, 

the  heat  evolved  is  equal  to  13,700  cal.  and  is  independent 
of  the  nature  of  the  base  and  acid  used  or  the  salt  formed, 
so  long  as  this  latter  at  that  dilution  fulfills  the  condition 

^  =  i. 

These  facts,  taken  in  connection  with  those  mentioned 
above  (pp.  76-90)  and  the  conclusions  arrived  at  there, 
are  not  so  startling  as  one  might  imagine  at  first  glance. 
Since  for  the  acid  and  base  we  have  the  relation 

/Wid 


SOLUTIONS.  91 

and,  since  the  salt  is  observed  to  have  a  formula  weight 
(by  definition)  equal  to  one-half  the  generally  accepted 
formula  weight,  i.e.,  is  completely  ionized  according  to 
all  the  possible  methods  of  measurement,  it  is  quite  cer- 
tain that  it  is  made  up  of  the  substances  previously  com- 
posing the  acid  and  base  in  the  same  state  as  that  in 
which  they  existed  in  them.  In  other  words,  expressing 
the  chemical  equation  in  accord  with  the  experimental 
facts  above,  we  have 


where  n  represents  the  number  of  moles  of  water 
present  in  the  system  before  the  reaction. 

Since  the  conductivity  shows  the  constituents  of  the 
salt  (the  two  kinds,  +  and  —  ,  of  ionized  matter)  to  be 
present  in  the  same  form  they  were  in  originally,  the 
only  portion  of  the  reaction  which  could  possibly  involve 
heat  is  the  formation  of  water  from  ionized  hydrogen  (H") 
and  ionized  hydroxyl  (OH').  As  we  know  that  hydrogen 
and  hydroxyl  in  the  ionized  state  can  exist  together  to 
but  an  infinitesimal  extent  (for  pure  water  conducts  only 
very  slightly),  the  following  conclusion  is  certainly  justified  : 
When  an  acid  unites  with  a  base  (at  any  rate  in  the  con- 
dition in  which  we  have  assumed  them)  the  cause  of  the 
reaction  is  the  inability  o)  ionized  hydrogen  to  exist  in  the 
presence  of  ionized  hydroxyl  beyond  an  exceedingly  small 
amount,  and  the  heat  of  the  neutralization  (for  this  case) 
is  that  heat  which  is  evolved  during  the  formation  of  water 
from  its  constituents  in  the  ionized  state  in  this  way,  i.e., 
13,700  cols,  for  each  mole  of  H'  and  OH'  (by  definition) 
forming  one  mole  of  H2O. 

By  a  method  which  we  shall  consider  later  (Chapter  VI) 


92  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

it  is  possible  not  only  to  show  the  presence  of,  but  to 
calculate  accurately,  the  heat  involved  in  the  ionization 
of  a  substance.  When  the  acid  and  salt  are  completely 
ionized,  for  example,  and  the  base  but  slightly,  it  is  pos- 
sible to  show  just  how  much  extra  heat  (either  positive 
or  negative)  is  involved  by  the  further  ionization  of  the 
base.  For  the  partly  ionized  base  must  increase  in 
ionization  as  its  ionized  OH'  is  used  up,  since  the  more 
dilute  the  solution  of  the  base  the  greater,  up  to  a  certain 
point,  is  its  ionization. 

If  both  the  acid  and  the  base  are  but  partly  ionized  the 
result  will  differ  still  more,  for  heat  will  be  absorbed  or 
evolved  by  the  further  ionization  of  both  of  these.  In 
general,  we  shall  have,  then,  if  the  salt,  also,  is  not 
completely  ionized,  i.e.,  if  more  heat  is  liberated  by  its 
undissociated  product  being  formed, 


-0:3)  - 
where 


a  1=  ionization  of  the  acid,  wi=its  heat  of  ionization; 
a2  =        "         "  the  base,  w2=  "    "     " 
«3=        "         "  the  salt,    w3=  "    "     " 
#  =heat  of  association  of  i  mole  of  ionized  H*  with 

i   mole   of   ionized  OH'  to  form  i   mole  of 

un-ionized  H2O; 

i.e.,  the  heat  generated  by  the  neutralization  of  an  acid 
by  a  base  is  equal,  for  each  mole  of  water  formed,  to  13,700 
col.  plus  the  product  of  the  heat  of  ionization  of  the  salt 
into  its  un-ionized  portion  minus  the  same  products  for 
the  acid  and  base. 


SOLUTIONS.  93 

Naturally,  the  negative  value  of  the  heat  of  association 
of  ionized  H*  with  ionized  OH'  is  the  heat  of  ionization  of 
water,  i.e.,  the  heat  necessary  to  form  i  mole  of  ionized  H* 
and  i  mole  of  ionized  OH'  from  water. 

Later  we  shall  consider  this  relation  more  in  detail, 
i.e.,  after  we  have  studied  the  method  to  be  used  for 
the  measurement  of  the  heat  of  dissociation. 

It  is  obvious  from  the  above  that  the  thermal  prop- 
erties of  electrolytes  are  additive  when  they  are  in  such 

a  dilution  that  they  fulfill  the  condition  — -  =  i ;  and  when 

r*» 

not  in  this  condition  the  change  in  the  thermal  effect 
depends  upon  the  amount  of  heat  involved  in  causing 
them  to  attain  this  state. 

When  a  precipitate  is  formed  in  such  a  solution  (i.e., 
when  a  chemical  reaction  takes  place,  which  was  excluded 
above)  it  is  often  possible  to  find  its  heat  of  formation 
just  as  we  found  that  of  water- above.  An  example  of 
this  is  the  following : 

Ag'  Aq  +  NO3'Aq  +  Na' Aq  +  Cl'Aq 

=  AgClAq+Na'Aq+NO3'Aq  +  i5,8oo  cals. 

or 

Ag' Aq  +  Cl'Aq  =  AgClAq  + 1 5,800  cals. 

i.e.,  when  i  mole  of  un-ionized  AgCl  is  formed  in  a  solu- 
tion from  the  ionized  silver  and  ionized  chlorine  15,800 
calories  are  evolved.  Conversely,  if  i  mole  of  AgCl  were 
dissolved,  this  amount  of  heat  would  be  absorbed,  i.e., 
the  heat  of  solution  of  a  substance  is  equal  to  the  nega- 
tive value  of  the  heat  of  precipitation. 


94  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

Although  it  is  not  always  possible,  we  can  find  the  heat 
of  formation  in  solution  in  still  another  way.  The  princi- 
ple of  this  is  as  follows:  By  electrical  measurements  it 
has  been  possible  to  find  the  amount  of  heat  involved 
when  2  grams  of  gaseous  hydrogen  form  2  grams  of 
ionized  hydrogen  in  solution.  This  value  is  approxi- 
mately equal  to  4  J,*  but  since  there  is  some  uncertainty 
about  its  exact  value,  it  is  usual  to  assume  it  equal  to 
zero.  Later,  then,  when  this  value  has  been  accurately 
determined,  the  results  found  in  this  way  can  be 
readily  recalculated.  From  this  value,  by  dissolving  a 
metal  in  a  completely  ionized  acid,  i.e.,  by  the  substi- 
tution of  metal  in  the  ionized  state  for  the  hydrogen, 
which  is  evolved  as  a  gas  from  that  state,  we  can  observe 
directly  the  heat  of  formation  of  the  ionized  metal  from 
massive  metal.  By  then  determining  the  heat  of  solu- 
tion of  a  completely  ionized  salt  of  this  metal,  the  heat 
due  to  the  negative  radical  in  the  ionized  state  can  be 
determined  readily,  for  the  heat  of  solution  of  the  salt 
is  equal  to  the  sum  of  the  heats  of  ionization  of  the  con- 
stituents, of  which  we  assume  that  of  hydrogen  to  be 
zero. 

In  this  way  the  table  given  below  has  been  prepared 
by  Ostwald.  In  order  to  find  the  heat  of  formation  of  a 
salt  it  is  only  necessary  to  obtain  the  sum  of  the  heats  due 
to  the  kinds  of  ionized  matter  into  which  it  decomposes, 
taking  into  account  the  valence  of  the  ionized  matter  as 
indicated  by  the  dots  for  the  electro-positive  and  the 
accents  for  the  electro-negative  substances. 


*  One  joule  (j)=io7  ergs=o.239i  cal.,  i.e.,  i  cal.  =  4.i83  j.  A  unit 
a  thousand  times  as  great  as  j  is  designated  by  J,  and  we  find  I  J=  239.1 
cal.=  io10  ergs,  i.e.,  i  cal.  =  0.004183  J. 


SOLUTIONS. 

95 

Cathion  Matter. 

J  =  joules  X  i  o3. 

Anion  Matter  of 

J  =  joules  Xio3. 

Hydrogen 

IT 

+     o 

Hydrochloric  acid 

Cl'         +    164 

Potassium 
Sodium 

K- 

Na' 

+  259 

+  240 

Hypochlorous  acid 
Chloric  acid 

CIO'     +  109 
CIO/    +     98 

Lithium 

Li' 

+  263 

Perchloric  acid 

CIO/    -   162 

Rubidium 

RV 

+  262 

Hydrobromic  acid 

Br7        +    118 

Ammonium 

NH/ 

+  137 

Bromic  acid 

BrO/    +     47 

Hydroxylamine 

NH,O 

+  i57 

Hydriodic  acid 

I'          +     55 

Magnesium 

Mg" 

+  456 

lodic  acid 

IO/      +  234 

Calcium 

Ca" 

+  45»(?) 

Periodic  acid 

IO/      +   195 

Strontium 

Sr" 

+  501 

Hydrosulphuric  acid 

S"         -     53 

Aluminium 

AT" 

+  506 

HS'       +       5 

Manganese 
Iron 

Mn" 
Fe" 

+  210 

+  93 

Thiosulphuric  acid 
Dithionic  acid 

S2O/'    +581 
S20/'    +1166 

Fe- 

—  39 

Tetrathionic  acid 

S40/'    +1093 

Cobalt 

Co" 

+   7i 

Sulphurous  acid 

SO/'     +  633 

Nickel 

Ni" 

+  67 

Sulphuric  acid 

SO/'     +897 

Zinc 

Zn" 

+  147 

Hydrogen  selenide 

Se"       -   149 

Cadmium 

Cd" 

+  77 

Selenious  acid 

SeO/'  +  501 

Copper 

Cu" 

-  66 

Selenic  acid 

SeO/'  +  607 

Or 

-   67(?) 

Hydrogen  telluride 

Te"      -    146 

Mercury 

Kg' 

-  85 

Tellurous  acid 

TeO/'  +   323 

Silver 

Ag- 

—  106 

Telluric  acid 

TeO/'  +  412 

Thallium 

Tl- 

+     7 

Nitrous  acid 

NO/     +   113 

Lead 

Pb" 

+        2 

Nitric  acid 

NO/     +   205 

Tin 

Sn- 

+     14 

Phosphorous  acid 

HPO/  +   603 

Phosphoric  acid 

PO/"  +1246 

HPO/'  -f  1277 

Arsenic  acid 

AsO/"+  900 

Hydroxyl 

OH'      +228 

Carbonic  acid 

HCO/+   683 

CO/'    +  674 

These  numbers  hold  only  for  the  case  that  the  ionized 
matter  is  in  very  dilute  solution,  i.e.,  Aq  should  be  added 
to  the  symbol  of  each  kind.  For  stronger  solutions,  in 
which  the  ionization  is  not  complete,  other  amounts  of 
heat  are  involved  which,  unless  allowed  for,  will  lead  to 
incorrect  results. 

The  equations 


J 


and 


96  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

mean  that  by  the  transformation  of  the  accepted  formula 
weight  of  metallic  sodium  into  the  ionized  state  240  J  are 
evolved,  and  for  the  change  of  the  formula  weight  of 
chlorine  gas  into  two  formula  weights  of  ionized  chlorine 
(by  definition,  p.  84)  2X164]  are  liberated. 


CHAPTER  V. 

CHEMICAL  MECHANICS. 

The  law  of  mass  action.  —  In  considering  such  a  reaction 


as 


or  any  other  reversible,  gaseous  process  which  finally 
attains  a  state  of  equilibrium,  the  question  naturally 
arises,  Iri  which  direction,  and  to  what  extent,  will 
the  reaction  go,  when  we  start,  for  instance,  with  a 
certain  amount  of  the  three  gaseous  constituents,  HI,  I, 
andH? 

From  the  purely  chemical  point  of  view  the  above 
equation  simply  provides  that  if  we  start  with  i  mole 
of  hydrogen  and  i  mole  of  iodine,  and  if  these  unite 
completely,  2  moles  of  hydriodic  acid  gas  will  be  formed; 
or  if  we  start  with  2  moles  of  hydriodic  acid  gas,  and 
this  is  completely  decomposed,  we  shall  obtain  i  mole 
each  of  hydrogen  and  iodine.  As  to  what  portion  of 
the  hydrogen  and  iodine  will  unite  to  form  hydriodic 
acid,  or  what  portion  of  a  definite  original  amount  of 
hydriodic  acid  will  decompose  to  form  hydrogen  and 
iodine,  or  what  will  take  place  if  all  three  substances 
are  mixed  together,  we  are  utterly  ignorant,  failing 
further  information  than  that  contained  in  the  chemical 
equation. 

97 


98  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

The  answers  to  these  questions  can  only  be  obtained 
by  the  application  of  a  very  general  law  which  was  first 
announced  by  Guldberg  and  Waage  in  1864.  The 
qualitative  form  of  this  law  of  mass  action  is  as  follows  * 
Chemical  action,  at  any  stage  of  the  process,  is  propor- 
tional to  the  active  masses  of  the  substances  present  at 
that  time,  i.e.,  to  the  amounts  of  each  present  in  the  unit 
of  volume. 

In  this  form,  however,  the  law  of  mass  action  is  of 
but  little  practical  use.  It  will  be  necessary,  then,  for 
us  to  derive  a  quantitative  expression  of  it,  and  thus 
to  obtain  it  in  such  a  form  that  it  may  be  applied 
to  our  needs  in  answering  questions  similar  to  those 
above. 

Imagine  a  reaction  of  the  type 

+  n2A  2  +±  n\A  /  +  nJA  2' 


having  taken  place  in  a  closed  vessel  and  to  have  attained 
a  state  of  equilibrium  in  which  we  have  the  partial  pres- 
sures pi,  p2,  pi  and  p2f. 

Assume,  further,  that  it  is  possible  to  insert  each  of 
the  substances  on  the  left  against  its  gaseous  or  osmotic 
pressure,  pi,  p2,  and  to  remove  each  of  the  products, 
as  they  are  formed,  from  the  gaseous  or  osmotic 
pressure  pi,  p2  to  the  original  external  pressure  po, 
and  that  this  insertion  and  removal  is  isothermal  and 
reversible. 

Since  by  such  a  series  of  operations  we  would  do 
work  on  one  side  (  —  ),  and  obtain  work  (  +  )  from  the 
other,  the  sum  of  the  two  amounts  (regarding  the  signs) 
would  give  us  an  expression  for  the  work  (+  or  —  )  which 
is  done  by  the  system  itself  during  the  transformation, 


CHEMICAL  MECHANICS.  99 

at  constant  temperature,  of  n\  moles  of  A\  and  n2  moles 
of  A 2  to  n\'  moles  of  A\  and  n2  moles  of  A2,  the  initial 
and  final  pressures  being  the  same,  viz.,  pQ.  And  this 
in  its  turn  would  lead  to  the  expression  of  the  quan- 
titative relation  existing  between  the  active  masses  of 
the  constituents  at  equilibrium,  i.e.,  to  the  relation  we 
seek. 

Since  the  work  required   to  change  the  osmotic   or 
gaseous  pressure  of  i  mole  of  substance  from  p0  to  pi  is 

given  by  the  expression  RT\og—  ,*  that  for  n\  moles 

Pi 
will  be  n>iRT  log.  —  .     For  n2  moles  of  A 2  we  have,  then, 

Po 

p2 
the  corresponding  expression  n2RT\oge  — .    The  sum  of 

these  two  terms  is  the  work  done,  i.e.  lost,  by  us  in  the 
process.  The  gain  of  work  for  us  then  for  this  stage  is 


By  the  removal,  as  they  are  formed,  of  n\'  moles  of  A\ 
and  n2  moles  of  A2,  the  amount  of  work  (a  gain  for  us) 
is 


In  total,  then,  our  gain  in  work  in  transforming  n\ 
moles  of  AI  and  n2  moles  of  A2  into  n\    moles  of  A\ 


loo   PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS 

and  n2   moles  of  A2   at  constant  temperature,  the  initial 
and  final  pressure  being  p^  is 


or 

W  =  RT(ni  log,  po+n2  log,  po-ni'  log,  pQ-n2'  log,  p0) 
I    +RT(ni'  log,  pi'  +n2'  log,  #2'  -»i  log,  #1  -»2  log,  #2). 

But,  as  we  simply  wish  to  get  the  relation  which  depends 
upon  the  pressures  in  the  reaction  at  equilibrium,  and 
the  pressure  po  has  nothing  to  do  with  this,  we  can  assume 
po  to  be  i,  and  obtain,  since  the  first  term  is  equal  to  zero, 

W  =  RT(m'  log,  pi'+nj  log,  p2'-ni  log,  pv  -n2  log,  p2). 

As  the  processes  of  insertion  and  removal  are  assumed 
to  be  isothermal  and  reversible,  this  work,  W,  must  be 
the  maximum  work  which  can  be  done  by  the  reaction,  and 
hence  must  be  a  constant  at  any  one  temperature. 

We  have,  then, 


anc  since  if  the  logarithm  is  a  constant  the  expression 
itself  must  be  a  constant,  and  since  T  and  R  are  also 
constants, 


_ 

Constant  =  K  = 


or,  since  pressure  and  concentration  are  proportional, 


CHEMICAL  MECHANICS.  101 

when  the  values  of  K  and  K'  may  or  may  not  be  alike, 
according  as  we  have  the  same  number  of  moles  on 
each  side  of  the  chemical  equation,  or  a  different  number. 

The  constant  (K  or  K')  is  known  as  the  constant  of 
equilibrium. 

We  may  express  the  law  of  mass  action  as  follows,  then : 
At  equilibrium  the  product  0}  the  pressures  (concentra- 
tions) of  the  substances  on  the  right  (final  ones),  each 
raised  to  a  power  equal  to  the  number  of  formula  weights 
reacting,  divided  by  the  product  of  the  pressures  (concen- 
trations) of  the  substances  on  the  left  (initial  ones),  each 
raised  to  a  corresponding  power,  is  a  constant  for  any 
one  reaction  at  any  definite  temperature* 

The  variation  of  this  constant,  K  or  K',  with  the  tem- 
perature is  to  be  considered  later,  after  we  have  studied 
the  application  of  this  most  important  and  general  law. 

Equilibrium  in  gaseous  systems. — For  gases  we  can 
most  conveniently  use  the  form  of  the  law  of  mass  action 
which  refers  to  partial  pressures.  We  have,  then,  for 
the  equilibrium  of  a  gaseous  chemical  system 


An  illustration  of  the  application  of  this  formula  is 
'given  by  the  gaseous  reaction 

2HI=H2+I2. 


*  In  case  the  reader  cannot  follow  this  derivation  he  should  at  any 
rate  memorize  this  law  and  thoroughly  understand  the  meaning  of  the 
two  mathematical  forms  of  it. 


102  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

The  reaction,  whatever  the  original  amounts,  will  only 
progress  until  the  pressure  of  H  is  pi  that  of  /  is  p2,  and 
that  of  HI  is  p,  where 


K  being  a  constant  depending  only  upon  the  temperature 
and  the  nature  of  the  reaction.  This  reaction  has  been 
studied  by  Bodenstein,  who  found  by  experiment  that 
K  at  444°  C.  is  equal  to  0.02012. 

If  we  heat  hydriodic  acid,  then,  to  this  temperature 
it  is  possible  to  calculate,  from  the  original  quantity 
the  amounts  of  hydrogen,  iodine,  and  undecomposed 
hydriodic  acid  present  at  equilibrium  when  the  tempera- 
ture is  440°,  for  under  these  conditions  pn=pi- 

.If  we  have  a  mixture  of  H,  I,  and  HI,  of  which  the 
partial  pressures  are  respectively  a,  b,  and  d,  and  wish 
to  find  in  which  direction  and  to  what  extent  the  reaction 
will  go  at  440°,  we  proceed  as  follows:  Let  x  represent 
the  unknown  partial  pressure  of  H  lost  to  form  HI, 
where  x  may  be  positive  or  negative;  then,  according 
to  the  chemical  equation  2HI=H2+l2>  we  shall  have 
at  equilibrium,  as  the  partial  pressure  of  HI, 


as  that  of  H  uncombined, 


CHEMICAL  MECHANICS.  103 

and  of  /  uncombined, 


And  x  must  have  such  a  value  (i.e.,  the  reaction  must 
go  so  far)  that  at  equilibrium  it  will  just  satisfy  the  equa- 
tion of  the  law  of  mass  action  for  the  reaction  at  440°, 


(a-x)(b-x) 

K=  0.0201 2  = 


Knowing  the  values  of  a,  b,  and  d,  in  this  system  it  is  then 
possible  to  solve  the  equation  for  #,  and  to  find  how  and 
how  far  the  reaction  will  go.  For  example,  in  this  case, 
if  x  is  found  to  be  positive  in  value,  the  reaction  will  go 
toward  the  left,  as  we  have  assumed;  if  negative,  in  the 
opposite  direction. 

There  is  one  thing  to  be  said  of  the  solution  of  such 
equations.  There  are  two  possible  values  of  x\  which 
is  to  be  taken  ?  It  will  be  found  in  this  case,  as,  indeed, 
in  all  others,  that  only  one  value  is  hi  accord  with  the 
existing  data,  so  that  it  alone  could  be  taken.  For 
instance,  if  the  positive  value  of  x  is  larger  than  a  or  b, 
it  would  lead  to  an  absurdity,  for  it  would  show  a  nega- 
tive value  for  H  or  /,  and  the  other  value  is  the  correct 
one.  In  cases  of  equations  of  a  higher  degree,  where 
more  than  two  roots  exist,  this  same  rule  is  to  be  fol- 
lowed. A  possible  case  here  is  to  have  two  values  of 
the  same  sign,  but  one  smaller  than  the  other.  There 
can  be  no  question  in  such  a  case,  however,  for  if  the 
reaction  would  be  in  equilibrium  after  the  smaller  change 
had  occurred,  it  could  not  go  out  of  this  state  to  attain 
the  equilibrium  shown  by  the  greater  value,  hence  the 
lower  value  is  to  be  taken  as  the  correct  one. 


104  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

Here  we  have  used  the  partial- pressure  form  of  the 
law  of  mass  action;  we  could  use  the  other  just  as  well, 
however,  for  it  will  be  observed  that  the  constant  factor 
which  would  transform  pressures  to  concentrations, 

P 

c  — =r,    is   eliminated,    since  we   have   the   same 

22-4X 

273 

number  of  formula  weights  (2)  on  each  side  of  the  equa- 
tion 2HI=H2+l2>  For  this  reason  the  constant,  K, 
for  this  reaction,  as  for  all  others  with  the  same  number  of 
formula  weights  on  both  sides,  has  the  same  value  for 
concentrations,  pressures,  volumes  under  standard  con- 
ditions, or  any  other  terms  in  which  the  amounts  may 
be  expressed,  so  long  as  they  are  proportional  to  the 
formula  weights. 

A  further  effect  of  this  condition  of  equal  volume  on 
the  two  sides,  is  that  the  direction  of  the  reaction  is  per- 
fectly independent  of  pressure  (Le  Chatelier's  theorem, 
p.  42) ;  and  Lemoine  has  shown  this  to  be  true  for  the 
decomposition  of  HI  for  pressures  ranging  from  0.2  to 
4.5  atmospheres. 

In  using  concentrations  in  place  of  partial  pressures 
it  is  always  to  be  remembered  that  the  concentration 
(i.e.,  moles  per  liter)  is  the  actual  number  of  moles  present 
divided  by  the  total  volume  (see  pp.  32-33).  An  exam- 
ple will  perhaps  make  this  clearer.  In  the  reaction 
A  =  2B  +  D,  at  equilibrium,  we  have  o.i  mole  of  A,  0.3 
of  B,  and  .05  of  D  in  10  liters  at  atmospheric  pressure 
and  o°.  Starting  with  0.5  mole  of  A,  o.i  of  B,  and  0.4  of 
D)  in  22.4  liters,  find  direction  and  extent  of  the  reaction. 

Here  we  must  first  find  the  constant  of  equilibrium 
for  the  data  given  at  equilibrium.  Since  we  have  o.i 
mole  of  A  in  10  liters,  the  concentration  of  A,  at  equilib- 


CHEMICAL  MECHANICS.  105 


rium,  is  —  ,  of  B  —  ,  and  of  D  -  ; 
10  10  10 


/0.3W0.05\ 

fe/ 


hence 

\  I0 

Assuming  that  x  moles  of  A  are  formed  by  the  reac- 
tion, the  final  volume  will  be  [(o.$+x)  +(o.i  —  2x) 
+  (0.4— x)]22>4  liters,  the  temperature  remaining  con- 
stant at  o°,  i.e.,  (1-2^)22.4  liters.  The  concentrations 

at  equilibrium,  then,  will  be  7 — -^-r moles  per  liter 

(i- 2^)22.4 

of  A,  - — '- — r of  By  and  - — '- — r     -  of  D\  hence  the 

(1—2^)22.4  (1-2^)22.4 

value  of  K,  as  found  above,  is  to  be  equated  to  these 
values  in  the  following  way: 


/      O.I—  2X     \2/       0.4  — #       \ 
\(l-2X)22.4/    \(l-2>r)22.4/ 

°"5+^        \ 
-2X)22.4/ 


K 


and  the  sign  of  x  will  show  the  direction  of  the  reaction, 
and  the  numerical  value  its  extent  (p.  103). 

Since  according  to  the  law  of  mass  action  the  con- 
centration is  to  be  raised  to  a  power,  it  is  the  whole  frac- 
tion representing  it  which  is  to  be  so  treated,  i.e.,  the  num- 
ber of  moles  per  liter. 

When  applied  to  the  equilibrium  resulting  from  a 
gaseous  dissociation  the  constant  of  the  law  of  mass 
action  is  usually  designated  as  the  constant  of  dissocia- 


lo6  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

tion.  From  it,  it  is  possible,  just  as  above,  to  calculate 
the  degree  of  dissociation  from  a  certain  amount  of  the 
dissociating  substance,  or  how  much  of  the  products, 
when  present  alone,  or  with  the  substance,  will  unite 
to  form  the  substance  itself.  And,  conversely,  we  can 
calculate  K  for  each  of  the  substances  for  which  data 
was  given  on  pages  28  and  29;  and  the  values  will  be 
dependent  only  upon  the  temperature,  the  units  employed 
(i.e.,  c  or  p),  and  nature  of  the  substance. 

When,  as  is  the  case  here,  we  know  a,  the  degree  of 
dissociation  of  the  substance,  we  can  proceed  as  follows: 
For  the  reaction  PC75<=tPC/3  +  C72,  for  example,  we 

have  for  concentrations  the  relation  K=  -  -  -  '. 


Starting  with  i  mole  of  PC/s,  which,  if  undissociated, 
would  occupy  V  liters  at  atmospheric  pressure,  with  a 
as  the  degree  of  dissociation,  the  concentrations,  where 
V  is  the  final  volume,  i.e.,  (i+a)F/,  at  the  same  tem- 

V    _  xy 

perature  and  pressure,   are  as  follows:  For  PCl$    y    ? 

for  PC/a  -y,  and  for  chlorine  -77-,  all  at  equilibrium;  hence 
K,  which  we  wish  to  determine,  is  to  be  found  from 


_       a2 

= 


/.        a      a      i  —  a\ 
V'6"  F"XF""     F  /' 


At  250°  for  PC/s  a  =  0.8  (p.  29)  and,  since  the  pressure 
is  atmospheric,  i  mole  must  be  present  in  22.4  -  - 

/  O 

liters;    this  is  equal  to  the  term  V  above.     F,  then,  is 
equal  to  (i  +0.8)  (22.4  —  -  -  j,  and  we  have  as  the  dis- 


CHEMICAL   MECHANICS. 


sociation  constant  for  PC15  at  250° 

fo.8)2  OF 


From  the  value  thus  obtained  we  could  then  calculate 
the  direction  and  extent  of  the  reaction  at  250°  when 
we  start  with  definite  amounts  of  the  three  constituents, 
or  the  value  of  a  for  a  different  V,  i.e.,  when  the  pressure 
is  other  than  atmospheric.  This  value  will  only  hold 
for  the  temperature  of  250°,  however. 

A  physical  idea  of  the  dissociation  constant,  as  found 
for  concentrations,  can  be  obtained  by  aid  of  the  formula 

a2 
K=-( ry.     Assuming  that  a  is  equal  to  0.5,  i.e.,  that 

the  degree  of  dissociation  is  50%,  for  a  reaction  by  which 

i  mole  is  transformed  into  2,  we  find  that  K=  ,     .T,,  or 

(o-5)  F 

2K= -y.     The  dissociation  constant  of  such  a  reaction,  then, 

when  multiplied  by  2  is  equal  to  the  reciprocal  of  the  final 
volume  resulting  from  the  dissociation  of  i  mole  into  2  to  the 
extent  of' 50%.  This  volume  is  that  in  which  i  mole 
of  the  original  substance  must  be  placed  in  order  that 
at  that  temperature  it  may  dissociate  to  the  extent  of 

50%  into  two  others.     Since  -^,  the  reciprocal  of  the 

volume  produced  by  the  dissociation  of  i  mole,  is  equal 
to  C,  the  concentration  in  moles  per  liter,  we  also 
have  2K=C.  An  example  of  the  use  of  this  relation  is 
given  by  the  reaction  N2C>4+±NO2+NO2,  for  which 

a2 
K=  ,  _   ^=0.0138  (calculated  from  ^=183.69  mm., 


io8  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

^  =1.894,  ^  =  3.18,  i.e.  a  =  0.69  and  F=m,  all  at 
49°. 7).  Nitrogen  tetroxide,  then,  should  be  50%  disso- 
ciated at  a  concentration  of  2X0.0138  =  7^=0  moles  per 

liter,  or  at  a  dilution  of  i  mole  in  36.3  liters.  Experi- 
ment shows  that  at  this  temperature  01  =  0.493  at  the 
dilution  i  mole  in  40  liters,  which,  considering  the  single 
value  from  which  K  is  determined,  and  the  evident 
small  error  in  the  experimental  observations,  is  a  satis- 
factory agreement. 
A  similar  definition  could  also  be  deduced  for  the 

a3 
reaction  A  =  2B  +  D,  where  K=  .   _   ^2,  although  the 

above  simpler  one  suffices  for  a  physical  idea  of  the 
dissociation  constant. 

It  is  to  be  noted  here  that  the  product  of  the  substances 
on  the  right  of  the  equation  has  always  been  placed 
in  the  numerator  of  the  fraction  giving  the  value  of  K 
(p.  100).  This  arrangement,  of  course,  is  optional,  so 
long  as  it  is  retained  the  same.  In  the  one  case  the  value 
of  K  will  simply  be  the  reciprocal  of  that  of  the  other. 

In  speaking  of  dissociation  (p.  28)  it  was  mentioned 
that  the  addition  of  one  of  the  products  of  dissociation 
to  the  system,  or  their  previous  presence  over  the  disso- 
ciating body,  decreases  the  extent  of  the  dissociation. 
That  this  must  be  true  according  to  the  law  of  mass 
action  is  made  obvious  by  the  consideration  of  any  defi- 
nite case. 

Supposing,  for  example,  in  the  case  of  phosphorus 
pentachloride,  the  space  over  it  contains  chlorine  prior 

to   the  dissociation.     Since  the  ratio  — — ^^  must  be 

CPCls 

a  constant,    less   of   the   PCl$  will    dissociate,   for  less 


CHEMICAL  MECHANICS.  109 

of  it,  with  the  chlorine  already  present,  will  suffice  to 
cause  the  ratio  to  attain  the  value  it  must  possess  at 
that  temperature. 

Or  to  take  another  case,  suppose  that  o.i  mole  of  B 
(p.  104)  were  introduced  into  a  vacuum,  and  the  substance 
A  allowed  to  dissociate  into  this,  arrangement  being  made 
by  a  movable  piston,  for  example,  so  that  the  final  pres- 
sure would  be  atmospheric.  What  would  be  the  effect 
of  this  o.i  mole  of  B  upon  the  dissociation  of  A,  the  tem- 
perature being  o°?  If  the  amount  of  A  when  undisso- 
ciated  were  i  mole,  the  volume  occupied  by  it  and  the 
o.i  mole  of  B  would  be  i.i  (22.4).  Assume  the  disso- 
ciation to  give  rise  to  xf  moles  of  .5,  then  the  number  of 
moles  of  B  in  the  final  volume  would  be  (o.i  +#0i  and 

since  the  final  volume  will  be  o.i  +  ( i )  +-ocf,  i.e.  i.i 

\      2/     2 

times  the  original  one,  we  have 


O.I  +  :*/  2 


(I.  I+*022.4'  (LI 


(1.1+^)22.4' 

and 


K_  Ui-i+y 


(l.I  +^22.4 


no  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

where  the  value  of  K  was  found  above  (p.  105).  This 
yf  is  smaller  than  the  value  (x)  which  would  be  obtained 
from  i  mole  of  A  in  the  pure  state,  occupying  the  same 
volume  (i.e.,  (i.i+#0  22.4  litres)  at  the  same  tempera- 
ture. And  the  difference  between  them  is  the  depression 
of  the  dissociation,  in  terms  of  By  due  to  the  addition. 
Since  for  every  mole  of  A  lost,  two  of  B  are  formed, 

x— otf 

—  gives  the  decrease  of  the  dissociation  of  A    (in 

moles)  due  to  the  presence  of  the  o.i  mole  of  B. 

The  addition  of  an  indifferent  gas,  either  before  or 
after  the  dissociation,  to  a  system  composed  of  a  disso- 
ciating substance  and  its  products,  has  no  effect  upon 
the  degree  of  the  dissociation,  so  long  as  the  total  'volume 
is  unchanged,  for  then  the  partial  pressures  (and  con- 
centrations) remain  unaltered.  An  increase  of  volume, 
on  the  other  hand,  such  as  was  allowed  to  take  place 
above,  no  matter  what  its  cause,  results  in  an  increase  in 
the  degree  of  dissociation.  When  due  to  the  addition 
of  an  indifferent  gas,  this  is  the  only  effect,  and  the  nature 
of  the  gas  is  without  influence.  When  due  to  the  addi- 
tion of  one  of  the  constituents,  however,  it  is  partly  com- 
pensated by  the  depressing  effect  of  this  upon  the  disso- 
ciation,* according  to  the  law  of  mass  action,  and  it  is 
possible  to  cause  the  one  influence  to  just  compensate 
the  other,  so  that  no  change  in  the  dissociation  is  to  be 
observed  as  the  result  of  the  addition  with  an  increase 
of  volume. 

A  somewhat  more  complicated  case  of  the  application 

*  The  value  of  x'  above  is  larger  than  it  would  have  been  if  the  volume 
were  not  allowed  to  increase,  as  can  be  seen  by  substituting  22.4  in  the 
denominators  in  place  of  (1.1  +  ^)22.4.  The  very  expansion  of  volume, 
without  the  presence  of  B,  would  cause  the  dissociation  to  increase, 


CHEMICAL  MECHANICS.  HI 

of  the  law  of  mass  action  to  homogeneous  gaseous  sys- 
tems is  given  by  the  dissociation  of  carbon  dioxide,  ac- 
cording to  the  scheme 


If  at  equilibrium  at  any  definite  temperature  the  partial 
pressures  are  p  for  CO2,  p\  for  oxygen,  and  p2  for  CO 
(where  these  come  from  the  CO2),  then  for  that  tempera- 
ture 


and  if  oxygen  is  already  present  from  an  exterior  source 
to  the  pressure  a,  the  decrease  in  the  pressure  of  carbon 
monoxide  due  to  its  effect  upon  the  dissociation  can  be 
readily  calculated.  We  have,  then, 


(P  +  2X)2 

from  which  x  can  be  calculated  (p.  103).  p2— 2X  will 
then  give  the  partial  pressure  of  carbon  monoxide,  and 
p  +  2X  that  of  carbon  dioxide,  in  the  presence  of  a  of 
oxygen,  the  constant  remaining  as  above. 

For  carbon  dioxide  at  atmospheric  pressure  and  3000°, 
a  =  0.4,  i.e.,  0.5  of  the  total  pressure  (0.5  of  an  atmosphere) 
is  due  to  CO2,  0.33  to  CO,  and  0.17  to  oxygen,  conse- 
quently ^=0.704. 

The  constant  for  COg  may  also  have  a  different  value; 


H2  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 
it  is  that  which  is  obtained  from  the  formula 


and  is  equal  to  K'  =  —  -  —  ,   which,  with  the  above  data, 

leads  to  the  value  Kf  =  0.272. 

Naturally,  what  was  said  of  the  arrangement  of  the 
ratio  expressing  K  also  holds  here.  Either  constant  may 
be  used  for  this  temperature,  provided  we  always  use 
the  same  form  of  relation. 

This  is  also  true  for  the  reaction 


or 

which  may  be  written  by  the  a  formula  either  as 
2 


_      a  .        a     a      i  —  a 

= 


4a2         (.        /2«\2     i-a\ 
~i-a)7      V'6"  \V)       TV' 


or 


And  one  form  must  be  selected  and  retained. 

Thus  far  in  our  consideration  of  gaseous  equilibrium 
we  have  applied  the  law  of  mass  action  only  to  systems 
composed  exclusively  of  gases.  Other  systems  exist, 
however  (i.e.,  those  made  up  of  a  liquid  or  a  solid  which 
evolves  a  gas  or  a  number  of  gases),  to  which  the  law  of 
mass  action  can  be  applied  with  great  success.  And 
the  application  is  usually  much  simpler  in  such  cases, 
for  it  is  a  well-known  experimental  fact  that  a  liquid  or  a 
solid  gives  a  definite,  constant  gaseous  pressure  at  any 


CHEMICAL  MECHANICS.  113 

one  temperature,  and  that  the  amount  of  liquid  or  solid 
has  no  further  effect  so  long  as  it  produces  sufficient  gas 
to  cause  the  pressure  to  be  attained  in  that  volume.  A 
similar  action  is  observed  when  we  saturate  a  solution 
with  a  substance,  for  so  long  as  sufficient  solid  is  present 
to  saturate  the  solution  an  excess  will  not  supersaturate 
it.  The  active  mass  of  the  solid,  in  applying  the  law 
of  mass  action,  then  remains  constant  hi  value,  and  its 
effect  can  be  included  in  the  constant  of  equilibrium. 
Thus  in  the  reaction 

solid  CaCO3^  solid  CaO+gaseous  CO2, 

although  the  pressures  n\  and  7r2,  due  to  the  gaseous 
CaCO3  and  CaO,  p  being  that  of  CO2,  can  be  given  in 
the  equation 


a  constant  value  will  also  be  found  by  employing  the 
simpler  form 


For  since  at  any  one  temperature  n\  and  7r2  remain 
constant  p  must  also  be  constant,  i.e.  the  equilibrium  at 
any  one  temperature  depends  only  upon  the  pressure  of 
CO2  produced,  and  this  is  shown  to  be  true  by  experi- 
ment. 

In  the  same  way  it  has  been  shown  that  equilibrium  in 
the  reaction 

solid  NH4HS  < 


H4  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

is  present  when  K'n=pip2i   or,  in  the  simpler  form, 
when  K=pip2.    And  in  the  reaction 


solid  NH4OCONH2 


when  K'x=pi2p2,  or  when  K=pi2p2.  And  both  of  these 
results  are  confirmed  by  experiment. 

These  relations,  just  as  those  for  homogeneous  sys- 
tems, hold  also  after  the  addition  of  one  of  the  products 
of  dissociation,  or  when  one  of  the  products  is  initially 
present,  in  the  space  into  which  the  solid  is  to  sublime 
and  dissociate.  Contrary  to  the  case  of  homogeneous 
equilibrium  (p.  no),  however;  an  increase  of  volume  has 
no  effect  upon  the  equilibrium,  so  long  as  the  solid  (or 
liquid)  phase  is  present,  for  the  dissociation  pressure  is 
dependent,  in  such  a  system,  only  upon  the  tempera- 
ture and  nature  of  the  system. 

Equilibrium  in  liquid  systems.  —  The  reaction 

CH3COOH  +  C2H5OH  <±  CH3COOC2H5  +H2O, 

it  has  been  observed,  reaches  the  state  of  equilibrium 
when  we  have  present  J  mole  of  acid,  J  mole  of  alcohol, 
f  mole  of  ester,  and  f  mole  of  water,  provided  we  start 
with  i  mole  of  each  of  the  two  constituents  (either  acid 
and  alcohol,  or  ester  and  water). 

This  reaction  goes  very  slowly  at  ordinary  tempera- 
tures, but  when  it  reaches  the  above  final  state  it  remains 
in  it  indefinitely.  If  we  designate  by  v  the  volume  (in 
liters)  of  the  system,  and  start  with  i  mole  of  acid,  m 
moles  of  alcohol,  and  n  moles  of  ester  (or  water),  then 
in  the  state  of  equilibrium,  after  x  moles  of  alcohol  have 


CHEMICAL  MECHANICS.  115 

been  decomposed,  we  shall  have  moles  per  liter 

i-x            .   x  /     n+x\  ,  n  +  x 

of  alcohol, of  acid,  -  (  or I  of  ester,  and  - 

V  V  \          V    /  V 

f  or  - 1  of  water.    And,  applying  the  law  of  mass  action, 

we  obtain 

(n+x)x 


K 


(i-x)(m-xY 


In  the  special  case  of  equilibrium  above,   however, 
m=iy  n=o,  x=%;  hence 


This  value  of  K  is  one  of  the  few  which  are  practically 
independent  of  temperature.  At  10°  it  is  found  that 
65.2%  of  the  acid  and  alcohol  undergoes  change,  while 
at  220°  the  decomposition  is  but  66.5%. 

This  equation  has  been  tested  by  experiment  with  very 
satisfactory  results.  //  has  been  found,  also,  that  by  using 
a  large  amount  of  acetic  acid  to  a  small  amount  oj  alcohol, 
or  vice  versa,  the  formation  oj  ester  and  water  is  almost 
complete,  as  it  should  be  by  the  law  of  mass  action.  In 
the  same  way  a  large  amount  of  water  upon  a  small 
quantity  of  ester  causes  the  latter  to  be  almost  completely 
transformed.* 

Amylene  in  contact  with  acid  forms  an  ester,  accord- 
ing to  the  equation 


*  For  results  of  experiments,  see  "  Elements,"  p.  248. 


Ii6  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

If  x  is  the  amount  of  ester  formed  when  equilibrium  is 
established,  v  is  the  volume  of  the  system,  and  i  mole  of 
acid  is  used  for  a  moles  of  amylene,  at  equilibrium  we 

shall  have   moles  per  liter  of  amylene  and  -      -   of 

v  v 

'Y* 

acid  left,  while  —  moles  per  liter  of  the  ester  will  be  formed ; 


hence  K 


(a-x)(i-xY 

The  value  for  K  has  also  been  experimentally  determined 

in  this  case  and  was  found  to  be  -  — ,  the  value  of 

0.001205 

the   constant  in   the  reciprocal  form  K'  =  — 

ocv 

being  0.001205.* 

When  a  solid  goes  into  solution  its  action  is  apparently 
analogous  to  its  transformation  into  the  gaseous  state. 
A  saturated  solution,  thus,  in  contact  with  the  solid  at 
any  temperature  will  still  be  saturated.  We  have,  then, 
by  the  law  of  mass  action,  for  any  one  temperature, 


or 


where  c  is  the  concentration  of  solid  in  solution  and  varies 
with  the  temperature.  If  the  solid  in  going  into  solu- 
tion dissociates  (non-electrolytically)  into  other  substances, 
an  addition  of  one  of  these  should  cause  less  substance 

*  See  "Elements,  "p.  250. 


CHEMICAL   MECHANICS.  II? 

to  dissolve.     This  has  been  proven  by  Behrend  for  a 
solution  of  phenanthrene  picrate  in  absolute  alcohol,  in 
which  a  decomposition  into  phenanthrene  and  picric  acid 
takes  place  to  a  large  extent. 
By  the  law  of  mass  action 


where  c  =  undissociated  phenanthrene  picrate,  Ci=free 
picric  acid,  and  £2  =  free  phenanthrene  —  all  expressed  in 
moles  per  liter.  For  any  one  temperature  c  must  be 
constant,  since  the  solution  is  saturated;  hence 

=  constant. 


It  is  obvious  that  this  will  also  enable  us  to  find  the 
conditions  of  the  equilibrium  attained  when  a  substance 
is  distributed  between  two  non-miscible  solvents  (p.  74). 
When  the  formula  weight  is  the  same  in  both  solvents, 

€2  1^2  W\ 

we  shall  have  K  =  —  ,  where,  since  ^2  =  irr>  and  £i=~77> 
c\  MM 

I  ^2\ 

the  ratio  of  the  concentrations  in  grams  (  —  )  per  liter 

must  also  remain  constant,  independent  of  the  original 
concentration  of  substance. 

In  case  the  formula  weight  in  one  solvent  is  n  times 
that  in  the  other  (Ai=nA2),  we  shall  have  Kc\  =c2n,  where 
Ci  would  be  equal  to  nc2  -if  all  A  i  were  transformed  into 


A2.     Since  Ci  =  irr>   CI  =  ^TI   and   Mi=nM2,    Kci=c2n 

MI  M2 

A/2n  W-21 

can  also  be  written  in  the  form  ~^-K  =  --  ,  where  Wi 

MI  Wi 


Ii8  PHYSICAL  CHBM/SfRY  FOR  ELECTRICAL  ENGINEERS. 
and  w2  SLTQ  the  weights  in  grams  in  a  certain  volume. 

But,  as  -r-T"  is  a  constant  so  long  as  the  ratio  of  the 

Mi 

formula  weights  does  not  change  with  the  dilution,  we 

M2n 

may  include  the  constants  ~^j—  and  K  in  a  new  constant, 

MI 

and  say,  when  the  formula  weight  in  one  solvent  is  always 
n  times  that  in  the  other  (at  the  dilutions  in  question),  the 

W2n 

ratio  of  distribution  in  the  form  -  will  remain  constant, 

Wi 

independent  oj  the  original  dilution.    An  illustration  of 
this  is  given  by  the  distribution  of  benzoic  acid  between 

benzene    (w±)  and    water  (w2),  the  values    of  -—  ,  for 
various  original  concentrations,  being  0.062,  0.048,  and 

Wo 

0.030,  while  those  of      ,  —  (which  must  be  constant  if 
w22 


\ 
is),  for  the  same  dilutions,  are  0.0305,  0.0304,  and 

/ 

0.0293,  the  differences  lying  well  within  the  experimental 
error.  The  formula  weight  of  benzoic  acid  in  benzene, 
then,  must  be  twice  that  in  water,  a  fact  which  has  been 
confirmed  by  aid  of  the  definition  of  formula  weight  in 
solution,  as  based  upon  freezing-point  depression. 

The  effect  of  temperature  upon  the  equilibrium- 
constant.  —  Wherever  a  change  in  temperature  changes 
the  equilibrium,  and  does  not  alter  the  nature  of  the 
equilibrium,  i.e.,  does  not  cause  the  disappearance  of 
any  of  the  constituents  of  the  previous  equilibrium,  it  is 
possible,  knowing  certain  factors,  to  find  the  effect  upon 
the  constant  of  equilibrium.  This  relation,  which  can 
be  readily  derived  from  the  law  of  mass  action  and  the 


CHEMICAL  MECHANICS.  119 

second  law  of  thermodynamics,*  we  shall  regard  as  given, 
and  simply  consider  its  typical  applications.  The  differ- 
ential form  of  the  equation  obtained  is 

—(log,  JQ  = 


where  q  is  the  heat  evolved  or  absorbed  as  heat  by  the 
reaction. 

To  integrate  this  expression  it  is  necessary  to  assume 
that  q  itself  is  independent  of  the  temperature.  This 
will  undoubtedly  be  practically  true  for  small  tempera- 
ture intervals;  for  larger  ones,  however,  we  must  be 
satisfied  to  obtain  q  as  the  value  for  the  temperature  which 
is  the  mean  of  the  two  extreme  temperatures. 

By  integration,  under  the  above  assumption,  we  find 


which,  using  ordinary  logarithms  and  solving  for  q,  is 
transformed  into 

2X2.306  (log  K'  -log  K)  TT 
q=  T-T'  -cals.,t 

where   2.306  is  the  reciprocal  of  the    modulus  of  the 
system  of  logarithms. 

This  formula  enables  us  to  determine  the  variation  of  the 
equilibrium  constant  K  with  the  temperature  (p.  101). 

*  See  "  Elements,"  pp.  452,  253. 

fThis  second  form    of    the    equation    is   to   be   preferred  for  the 
actual  calculations,  although  the  other  is  simpler  in  form. 


120  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

q  here  only  expresses  the  heat  of  the  reaction  as  it 
would  be  if  no  external  work  were  done  by  or  upon  the 
reaction.  The  allowance  for  this,  in  comparing  the 
results  of  the  formula  with  observed  results,  must  be 
made  in  each  case,  as  is  illustrated  in  the  applications 
given  below. 

One  consequence  of  this  formula  is  of  special  impor- 
tance. If  q  is  zero,  the  value  of  K  does  not  change  as 
the  result  of  a  variation  in  the  temperature.  Thus  the  re- 
action between  acid  and  alcohol,  mentioned  above  (p.  115), 
the  mutual  transformation  of  optical  isomers,  and  a 
number  of  others,  are  found  neither  to  absorb  nor  gen- 
erate heat,  nor  to  suffer  a  displacement  of  equilibrium, 
i.e.  a  change  in  the  value  of  K,  by  a  change  in  tempera- 
ture. 

We  shall  now  consider  the  method  of  applying  this 
equation  for  various  purposes  to  various  equilibria. 

Vaporization.  —  The  condition  regulating  the  equilib- 
rium between  a  liquid  and  its  vapor  is  the  pressure  or 
concentration  of  the  latter  (pp.  112-113),  and  this  de- 
pends upon  the  temperature.  We  have  then 

K-    L  JL 

~V~RT 

And  if  p  and  $  refer  to  the  two  temperatures  T  and 
Tr  we  may  write,  since  R  is  constant, 

P'   ,     P 


Regnault  found  for  water  that  at  ^  =  273,  ^  =  4.54 
mms.  of  Hg,  and  at  Tf  =  273  +  11.54,  pf  =  10.02  mms, 
of  Hg,  hence  q  is  equal  to  —  10100  cals.  for  i  mole  (18 


CHEMICAL   MECHANICS.  12  f 

grams)  of  water.  Direct  experiment  shows  this  value  to 
be  —10854  cals.,  but  here  external  work  to  the  amount 

fT+T'\ 

2!  -  )=557  cals.  (p.  44)  is  done  and,  consequently, 

must  be  subtracted  before  any  comparison  is  possible. 
We  find  in  this  way  that  <?=—  10297,  which  may  be 
regarded  as  in  satisfactory  agreement  with  —  10100. 

Dissociation  oj  solids.  —  When  a  solid  dissociates  into 
gases,  the  equilibrium  is  conditioned  by  the  concentra- 
tion or  pressure  of  the  latter.  In  the  reaction  solid 
NH4HS  =  #2S  +  A/7J3,  where  the  concentrations  of  #2.5 
and  NH%  are  Ci  and  c2,  that  of  the  gaseous  NH4HS  being 

i       pi 
negligibly    small    and    constant,  •  we    have    Ci=-y  =  j^, 


2 

£2  =  7^=7^;,    V    meaning    the   volume    which    contains 

i  mole. 
Neglecting  the  pressure  of  the  gaseous  NH4HS  it  is 

p 
obvious,  however,  that  —=^1=^2,  where  P  is  (practi- 

cally) the  total  pressure  at  a  constant  temperature.     We 

p2  p/2 

have  then  K=pip2  =  -y  and  K'  =  pi'p2'  =  T7>  where 


P'  refers  to  the  temperature  J1',  and  it  follows  that 

/       P'  P\ 

log,  K'  -  log,  K  =  2  (log  —  -  log  ^  j  = 


Since   at    273°  +  9°.5,    P=iy5    mms.    of  Hg,    and   at 
i,   Pr  =  5oi   mms.   of  Hg,  £=—21550,    while 


122  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

by  direct  experiment  the  value  is  —21639  (i-e->  —22800 

T+r>\ 
—  noi,  where  1161=4  -  )• 

The  general  form  of  the  equation  for  a  dissociation  of 
this  kind,  where  HI  moles  of  one  gas  and  n^  moles  of 
another,  etc.,  are  evolved,  is 


log,  £'-log,  £<==(»!  +»2+  .  .  .  )log.      -log, 


R\T     Tf° 

This  formula  may  also  be  applied  to  the  dissociation 
of  salts  containing  water  of  crystallization  into  gaseous 
water  and  the  dehydrated  or  partially  dehydrated  salt. 

Solution  o)  solids.  —  In  this  case  the  equilibrium  depends 
only  upon  the  concentration  of  solid  substance  in  the 
solution,  and  the  temperature  ;  we  have,  then, 

K=c, 

where  c  is  the  concentration  of  a  saturated  solution  at 
the  temperature  T.  If  cf  is  the  concentration  of  such 
a  solution  at  another  temperature  71',  then  it  is  possible 
for  us  to  calculate  the  heat  of  solution  of  the  solid,  the 
increase  of  volume  being  so  small  that  the  external  work 
is  practically  equal  to  zero. 

van't  Hoff  found  by  experiment  with  succinic  acid 
in  water  that  for  7^  =  273,  £  =  2.88  moles  per  liter;  and 
for  7^  =  273  +  8.5,  £'  =  4.22  moles  per  liter. 

For  i  mole,  then,  ^=6900  cals.,  while  Berthelot  found 
—  6700  cals.  by  direct  experiment. 

lonization  of  solids  in  solution.  —  If  a  substance  is  very 
slightly  soluble,  and  the  solution  consists  principally  of 
ionized  matter  with  very  little  undissociated  substance, 


CHEMICAL  MECHANICS.  123 

the  heat  of  dissociation  must  be  the  same  as  the  heat  of 
solution,  i.e.,  equal  to  the  negative  value  of  the  heat  of 
precipitation  from  the  two  kinds  of  ionized  matter. 
Thus  for  AgCl  we  have 


If  the  solubility  at  T  =  c,  and  at  T/  =  cft  in  moles  per 
liter,  then,  since  2  moles  of  the  ionized  matter  are  formed 
from  i  mole  of  the  salt,  we  have,  as  on  page  122, 


logX-log.«=4Vr     T 


For  T  =  273  +  20,  c=i.ioXio~5,  and  for  ^  =  273+30, 
c'  =  i.73Xio~5;  hence  q=  —15900=  —159^." 

For  the  negative  heat  of  precipitation  we  found  (p.  93) 
—  i$SK,  which  is  an  excellent  agreement.  We  have 
assumed  the  ionization  to  be  complete  here,  and  the 
fact  that  the  heat  results  agree  cannot  but  be  considered 
as  confirmatory  of  our  assumption. 

Dissociation  of  gaseous  bodies.  —  When  a  substance  A 
dissociates  according  to  the  scheme 


the  equation  of  equilibrium  is 


where   c,    c\,  c^  .  .  .  are    the    concentrations   of  A,  A\, 
AI,  .  .  .  in  moles  per  liter. 
If  the  mole   occupies  the  volume   V  at   T  and  the 


124  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

volume  F'  at  T1',  then  for  the  dissociation  of  N2O4  we 
have 

N2O4  +=*  2NO2, 
hence  (p.  112) 

K'  =  ,,  "'*          and    K 


(i  -OF'  (i-a)F 


a2          q/i       i 


Since  the  density  of  undissociated  N2O4,  from  its  for- 
mula weight  (96),  is  3.179  (based  upon  air),  a  for  any 

temperature  can  be  found  from  the  relation  a  = —  i. 

3-179 
Since  one  mole  of  gas  at  any  temperature  and  atmospheric 

pressure  occupies  0.0819  ^ liters  (from  — —  j,  the  volume 

occupied  by  i  mole,  which  has  undergone  dissociation  to 
the  extent  a  is  (i  +a)  0.0819  T  liters,  where  a  is  the 
dissociation  at  the  temperature  T. 

From  the  results  observed,  viz.,  7^  =  273  +  26.1,  ^  =  2.65; 
T'  =  273  +  111.3,  A'  =  1.65;  we  find  a  =  0.1986, 
F  =  359.i,  #'  =  0.9267  and  F'  =  67i,  hence  <?=— 12900 
cals.  Subtracting  the  work  necessary  for  expansion  * 
from  the  experimental  result,  we  find  q=  —12500  cals. 

Velocity  of  a  chemical  reaction. — As  up  to  the  present 
we  have  only  considered  the  equilibrium  which  is  at- 
tained after  the  reaction  has  come  to  rest,  we  must  now 

*  In  using  this  formula  it  is  to  be  remembered  that  g  is  the  heat 
appearing  as  heat,  and  does  not  include  that  used  in  overcoming  a 
pressure  when  expanding. 


CHEMICAL  MECHANICS.  125 

consider  very  briefly  the  laws  governing  its  progress 
toward  this  end.  Since  chemical  action  at  any  time, 
according  to  the  mass-law  (p.  98),  is  proportional,  for 
constant  temperature,  to  the  active  masses  of  the  sub- 
stances present,  i.e.  to  those  portions  which  are  free  to 
act,  then,  when  we  have  two  substances  reacting,  the 
concentrations  being  di  and  a2  moles  per  liter, 


dx 

— 


where  x  is  the  fraction  of  a  mole  of  each  which  decom- 
poses in  the  time  /.  The  term  k  in  this  equation  is  known 
as  the  speed  constant  of  the  reaction,  and  is  constant  at 
any  one  temperature  for  any  value  of  x  in  the  reaction 
in  question. 

Suppose  we  have  the  reversible  reaction 

Ai+A2<=±Ai'+A2', 

which  after  a  time  attains  a  state  of  equilibrium  in  which 
all  four  products  are  present.  The  relative  amounts  of 
these  are  dependent  upon  the  value  of  K  for  this  reaction 
at  this  temperature,  i.e.  are  fixed  by  the  relation 


If  we  start  with  a\  moles  of  A\  and  a2  moles  of  A2,  then 
dx 


126  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS 
But  if  we  start  with  a\   moles  of  A\   and  aj  of  A2, 


dx' 
where  ~r-  is  the  velocity  in  the  opposite  direction. 

Starting  with  the  substances  on  either  side,  then,  those 
on  the  other  will  exert  an  ever-increasing  influence  upon 
the  velocity  due  to  the  initial  substances,  and  this  veloc- 
ity must  decrease  continually.  Finally,  however,  equilib- 
rium will  be  attained  and  the  ratio  of  the  amounts  on  the 
two  sides  will  remain  constant,  i.e.,  the  reaction  as  a 
whole  will  cease,  and  any  motion  which  exists  will  be  so 
compensated  by  a  contrary  one  that  it  will  not  appear. 

Imagine  we  start  with  a\  moles  of  A  i  and  a2  moles  of 
A2.  The  total  velocity  due  to  these  at  any  one  time 
will  be 

dX    dx    dx' 

~==~~^  (ai~^(a2~^~   (x^y 


j  -y 

and  at  equilibrium,  i.e.  where  "jT^ 


k 
F 


i.e.,  the  equilibrium  constant,  K,  of  any  reversible  reaction 
is  equal  to  the  ratio  of  the  speed  constants  of  that  reaction 
for  the  two  directions. 

This  has  been  proven  to  be  true  for  a  number  of  cases. 
For  the  system  acid-alcohol  (p.  114)  it  was  found  for  a 


CHEMICAL   MECHANICS.  127 

certain  strength  acid  that  £  =  0.000238  and  kf  =  0.00081  5, 

k 
from  which  ^  =  -77=2.92,  while  direct  experiment  gave 

^  =  2.84. 

All  this  is  only  true,  however,  when  the  reaction  takes 
place  isothermally,  i.e.,  when  the  heat  liberated  or  absorbed 
is  removed  or  supplied  so  that  no  change  in  the  tempera- 
ture results,  jor  the  constants  k  and  kf  are  dependent  upon 
the  temperature. 

In  general  the  application  of  this  formula  is  very  much 
simplified  by  the  fact  that  most  reactions  are  almost 
complete  in  one  direction,  so  that  the  second  term  will 
be  so  small  that  it  may  be  neglected.  We  have,  under 
these  conditions, 

dx 


In  all  these  cases  the  values  on  the  right  are  obtained 
by  subtracting  the  loss  from  the  concentration  of  the 
original  substance  and  having  as  many  such  terms  as 
there  are  formula  weights  of  substance.  Thus  for  the 
reaction  A  =  2B+D  we  would  write 

dx 


This  is  simply  custom  (see  pp.  108  and  112),  for  we 
could  also  write  it 

~ 

and  although  the  kf  value  would  be  different,  it  would 
still  be  a  constant. 


128   PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

Reactions  of  the  first  order.  Catalysis.  —  For  con- 
venience we  shall  divide  all  reactions  into  orders.  Thus 
a  reaction  of  the  first  order  is  one  in  which  but  one  sub- 
stance suffers  a  change  in  concentration.  This  defini- 
tion is  to  be  further  restricted,  in  that  for  the  first  order 
it  is  necessary  that  the  equation  show  but  i  formula 
weight  of  substance  changing  its  concentration. 

Cane-sugar  in  water  solution  is  transformed  in  the 
presence  of  acids  almost  completely  into  dextrose  and 
laevulose;  it  is  inverted.  The  speed  of  the  reaction  is 
very  small  and  increases  with  the  amount  of  acid  added. 

The  progress  of  the  reaction  may  be  observed  by  aid 
of  the  polariscope.  The  uninverted  portion  revolves  the 
plane  of  polarization  to  the  right,  while  the  two  prod- 
ucts revolve  it  to  the  left. 

This  process  was  first  measured  by  Wilhelmy  (1850), 
and  it  has  played  an  important  r6le  in  the  history  of 
chemical  mechanics. 

The  process  follows  the  scheme 


whether  acid  is  used  or  not,  for  the  concentration  of 
the  latter  does  not  change  during  the  reaction.  Accord- 
ing to  the  law  of  mass  action  the  speed  is  proportional 
to  the  amounts  of  sugar  and  water.  The  latter,  how- 
ever, is  present  in  such  an  excess  that  its  action  may 
be  regarded  as  constant.  The  speed  of  reaction,  then, 
is  proportional  to  the  amount  of  sugar  present,  and  we 
have  a  reaction  of  the  first  order,  i.e.  the  relation  is 

doc 


CHEMICAL  MECHANICS.  129 

where  for  /  =  o,  x  =  o,  and  k  is  the  inversion  constant, 
which  depends  only  upon  the  temperature.  By  integra- 
tion this  becomes 


— log,  (a  —  x)  =  kt  +  constant, 
or,  since  for  t  =  o,  x  =  o, 

log,,  (a)  =  the  constant, 


in  other  words,  a  constant  fraction  of  the  total  amount 
of  sugar  is  inverted  in  each  unit  of  time. 

The  meaning  of  the  constant  k  in  words  is  as  follows : 
Its  reciprocal  value  multiplied  by  the  natural  logarithm 
oj  2  gives  the  time  in  minutes  which  is  necessary  for  the 
transformation  of  one  half  the  total  amount  oj  substance, 
provided  the  products  oj  the  reactions  are  removed  as  soon 
as  they  are  formed  and  the-  substances  replaced  as  they  are 

a 

used.  This  is  shown  by  the  substitution  of  -  for  x.  Fur- 
ther, for  all  reactions  of  the  first  order,  this  constant  k 
is  independent  of  the  original  concentration  of  the  sub- 
stance. 

Another  reaction  of  the  first  order  is  the  formation 
of  alcohol  and  acid  from  an  ester  with  water.  For 
example,  the  reaction 


130  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

And  here  just  as  above  we  find  a  constant  value  for  k 
in  the  formula. 

Both  these  reactions  are  examples  of  the  process  which 
is  known  as  catalysis;  that  is,  the  action  of  a  substance 
in  hastening  the  change  of  a  substance  without  at  the 
same  time  being  decomposed  itself.  In  both  these 
cases  the  hydrogen  in  the  ionized  state  (by  definition)  is 
the  catalytic  agent.  And  it  has  been  shown  for  more 
dilute  solutions  that  the  catalytic  action  is  strictly  pro- 
portional to  the  concentration  of  ionized  H',  as  found  (by 
definition)  by  aid  of  any  of  the  possible  methods.  In 
strong  acid  solutions,  however,  this  exact  proportionality 
does  not  obtain,  for  reasons  which  as  yet  are  not  thor- 
oughly understood.  At  any  rate,  the  method  as  it  stands 
enables  us  to  define  the  concentration  oj  ionized  H'  in 
the  weaker  solutions  of  acid. 

Reactions  of  the  second  order.  —  Here  two  formula 
weights  of  substance  suffer  a  change  in  concentration 
during  the  reaction,  i.e.,  the  constant  depends  upon  the 
concentration  of  two  substances.  We  have  then 

doc 


or     -  —  ^  [log^  (b  —  x)—  loge  (a  —  x)]  =  kt+  constant. 
And,  since  when  /  =  o,  x  =  o,  the  constant  is 


i  (a-x)b 

Le-  *=lo 


CHEMICAL  MECHANICS.  13* 

If  we  use  equivalent  amounts  of  the  two  substances,  then 
a  =  b,  and  we  have 


i        x 

or  k  =  — 


. 

t  (a—x)a 

In  a  reaction  of  the  second  order  k  is  inversely  proportional 
to  the  original  concentration. 
An  example  of  such  a  reaction  is 


=  CH3COONa  +  C2H5OH, 
or,  when  written  as  an  ionic  reaction, 

C2H3COOC2H5  +  Na-  +  OH' 

-  C2H300'  +  Na*  +  C2H5OH, 

for  a  base  acts  in  such  a  case  with  an  effect  proportional 
to  its  content  (by  definition)  of  ionized  OH'.  In  other 
words,  the  velocity  of  saponification  is  proportional  to 
the  concentration  of  ionized  OH'  present,  and  independent 
of  the  radical  from  which  it  is  split  off. 

For  weak  bases,  i.e.  those  which  are  but  slightly 
ionized  (NH4C1,  for  example,  where  a  in  a  W/4O  solu- 
tion is  0.0269,  as  compared  to  0.972  for  KC1),  the  salt 
formed,  by  its  depressing  effect  upon  the  ionization  of 
the  base  *  (p.  85),  for  C2H3O2Na  is  very  largely  ionized, 

*  It  is  a  well-known  experimental  fact  that  a  substance  which  is 
but  slightly  ionized  is  influenced  to  a  greater  extent  by  a  given  con- 
centration of  a  substance  containing  a  common  kind  of  ionized  matter, 
than  one  which  is  considerably  ionized.  For  the  calculation  of  this 
influence,  see  Chapter  VI- 


I32  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

is  found  to  influence  the  constancy  of  k,  but  this  can  be 
allowed  for  in  the  calculation,  as  we  shall  see  later. 
Thus  at  24°.  7  a  W/4O  .solution  of  KOH  gives  k  a  value 
of  6.41;  hence  k,  for  a  like  solution  of  any  other  base, 

must  be  equal  to  -      -  6.41,  where  a  is  the  percentage 

present  of  ionized  OH'  in  the  solution  of  that  base  in 
presence  of  the  salt  found.  This  a  is  a  value  that  we 
can  find  when  we  know  the  ionization  of  the  salt  formed. 
In  short,  then,  jrom  the  speed  of  saponification  of  a  base, 
knowing  that  produced  by  a  definite  concentration  of 
ionized  OH',  it  is  possible  to  define  and  determine  the 
concentration  of  ionized  OH'. 

Although  reactions  of  the  third  order  are  also  known, 
and  have  been  investigated,  their  importance  for  our 
purposes  is  very  slight,  and  consequently  they  need  not 
be  considered  here.  And  for  the  consideration  of  those 
reactions  which  are  incomplete,  as  well  as  of  those  which 
are  non-homogeneous,  and  the  discussion  of  the  effects 
of  temperature  upon  the  speed  of  a  reaction,  the  reader 
must  also  be  referred  elsewhere.* 

*  "  Elements,"  pp.  276-281. 


CHAPTER  VI. 

EQUILIBRIUM  IN  ELECTROLYTES. 

Organic  acids  and  bases.     The  Ostwald  dilution  law. — 

The  application  of  the  law  of  mass  action  to  gaseous 
equilibria,  as  well  as  to  those  existing  in  solutions  of 
non-electrolytes — in  short,  to  chemical  equilibria — has 
shown  that  it  is  a  general  law  of  nature,  holding  between 
very  wide  limits.  The  question  which  naturally  arises 
here,  then,  is,  Can  the  law  of  mass  action  also  be  applied 
to  those  equilibria  composed  of  the  ionized  and  unionized 
portions  of  a  substance  in  solution?  In  other  words, 
Is  the  law  of  mass  action  the  principle  governing  the 
amounts  of  these  portions  which  can  exist  together  in 
equilibrium  ?  It  is  the  purpose  of  this  chapter  to  answer 
this  question  so  far  as  is  possible  from  our  present  knowl- 
edge of  the  facts. 

The  conductivity  ratios  — ,  as  well  as  the  methods  for 

^oo 

determining  the  average  molecular  weight,  where  these 
can  be  carried  out  with  sufficient  accuracy,  show  that  a 
water  solution  of  acetic  acid  is  ionized  according  to  the 
following  scheme: 

CHaCOOH^CHsCOO'+H*. 

'33 


134  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS 

From  this  equation,  applying  the  law  of  mass-action, 
we  obtain 


CCH3COOH 

or  (see  page  106) 


which  is  known  as  the  Ostwald  dilution  law}  for  it  gives 
the  relation  of  ionization  to  dilution. 

Substituting  the  ratio  —  -  for  a    at  various  dilutions, 

r^oo 

Ostwald  found  K  to  be  constant,  with  a  value  at  25°  of 
i.8Xio~5.  From  the  value  of  K  at  any  temperature, 
then,  it  is  possible,  by  solving  for  a,  to  find  the  degree 
of  ionization  at  any  dilution  or  at  any  concentration  at 

that  temperature,  for  C  =    T.    We  find  thus  that 


K2V2    KV 


where  K  is  the  equilibrium,  dissociation,  or  ionization 
constant,  or  the  so-called  coefficient  oj  affinity,  of  the  acid. 

Further  than  this,  when  K  is  known  for  any  tem- 
perature it  is  possible  to  find  a,  at  any  dilution,  in  the 
presence  of  an  acid  or  salt  with  ionized  matter  in  common 
(H*  or  CH3COO')-  And,  just  as  in  the  case  of  gaseous 
dissociation,  we  always  find  a  smaller  dissociation  in  the 
presence  of  the  products  arising  from  an  exterior  source. 
Naturally  the  calculation  here  is  similar  to  the  other 
(pp.  108-109).  For  example,  we  have 


EQUILIBRIUM  IN  ELECTROLYTES.  135 

where  x  represents  the  concentration  of  ionized  H"  due 
to  other  substances,  CH-  and  £cH3coo'  are  the  concen- 
trations due  to  the  acid  in  the  absence  of  other  substances, 
and  y  is  the  concentration  of  each  (H'  and  CH3COO') 
lost,  i.e.,  uniting  to  form  un-ionized  CH3COOH,  as  the 
result  of  the  presence  of  x  moles  of  ionized  H*.  The 
concentrations  of  H*  and  CH3COO',  now  arising  from 
the  acid,  are  equal,  then,  to  (cn--y)  =  (ccH3coo'-y), 
and  the  un-ionized  acid  concentration  is  (cCH3cooH+y). 

Indeed,  everything  which  we  found  above  (pp.  IOI-IIG 
to  hold  true  for  gaseous  dissociation,  with  the  one  ex- 
ception mentioned  below,  also  holds  true  for  the  relation 
of  ionized  and  un-ionized  portions  of  a  substance,  of  the 
type  of  acetic  acid,  in  solution.  And,  as  a  rule,  the  re- 
sults are  simpler  to  calculate,  for  the  volume  change  of 
the  system  is  so  small  as  to  be  negligible. 

We  can  also  define  the  ionization  constant  in  terms 
similar  to  the  definition  of ,  the  gaseous  dissociation  con- 
stant (p.  107).  Thus  by  multiplying  K  for  acetic  acid 
by  2  we  obtain  0.000036  as  the  concentration  in  moles 
per  liter  of  a  solution  of  acetic  acid,  which  would  be  50% 
ionized,  i.e.,  a  solution  of  i  mole  of  acid  in  27777.5  liters 
of  water. 

All  organic  acids  when  treated  in  this  way  give  a  con- 

a2 

slant  value  jor  the  expression  _  .„.  This  is  the  dif- 
ference between  gaseous  and  electrolytic  dissociation 
equilibria,  at  least  when  the  latter  is  for  an  organic  acid. 
The  acid,  whether  it  be  mono,  di,  or  polybasic,  always 
ionizes  as  a  monobasic  acid  up  to  the  dilution  at  which 
a  =  0.5.  This  means  that,  assuming  the  acid  to  ionize 
simply  into  the  two  products  H*  and  the  negative  radical 
(i.e.,  for  the  calculation  of  fjLn),  which  may  also  contain 


136  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 


replaceable  hydrogen,  a  constant  is  obtained  so  long  as 
the  ionization  in  this  way  is  50%  or  less.  Beyond  that 
point  the  ionized  H*,  due  to  a  breaking  down  of  the 
negative  ionized  radical  containing  replaceable  hydrogen, 
is  great  enough  in  concentration  to  influence  the  constant, 
which  begins  to  vary.  Above  the  dilution  at  which 
a  =0.5  the  second  and  following  replaceable  hydrogens 
begin  to  appear  in  the  ionized  state,  and  must  be  taken 
into  account;  and  at  infinite  dilution,  if  it  were  possible 
to  attain  it,  the  polybasic  acid  would  be  composed  of 
all  the  replaceable  hydrogen  in  the  ionized  state,  together 
with  the  negative  radical,  without  replaceable  hydrogen, 
as  the  negatively  charged  ionized  matter.* 

The  ionization  constants  at  25°  C.  for  a  few  organic 
acids  are  given  below;  these  are  for  the  first  equivalent 
of  ionized  IT,  i.e.,  give  results  which  agree  with  experi- 
ment up  to  a  dilution  at  which  a  =  50%. 


IONIZATION  CONSTANTS  OF  ORGANIC  ACIDS  AT  25°  C. 


Propionic 1.34 

Isobutyric i .  44 

Capronic i .  45 

Butyric i .  49 

Valerianic i .  61 

Acetic i .  80 

Camphoric 2.25 

Anisic 3 . 20 

Phenylacrylic 3.55 

Succinic 6.65 

Lactic 13.8 

Glycollic 15.2 

Formic 21.4 


Malic 

Fumaric 

Tartaric 

Salicylic 

Orthophthalic 

Monochloracetic.  .. 

Malonic 

Maleic.  . 


KXio* 

39-5 
93 
97 
1 02 

121 

158 
II7O 


Dichloracetic 5140 

Oxalic ioo6o(±)f 

Trichloracetic 1 2 1000  (  ±  )  f 


*  For  the  calculation  of  the  amount  of  the  2d  and  3d  equivalents  of 
ionized  H'  at  any  dilution,  see  Smith,  Zeit.  f.  phys.  Chem.,  25,  144, 
and  193,  1898,  and  Wegscheider,  Sitzungsber.  d.  Akad.  d.  Wissenschaft., 
in,  441-510,  1902. 

t  These  two  acids  are  so  largely  dissociated  that  a  small  error  in  or 


EQUILIBRIUM  IN  ELECTROLYTES.  1  37 

The  value  of  K  at  18°  for  a  few  other  acids,  some  of 
which  are  inorganic,  but  characterized  by  their  small 
ionization,  and  by  obeying  the  law  of  mass  action,  are 
given  by  Walker  and  Cormack,*  the  value  of  acetic  acid 
being  given  for  comparison. 

Per  Cent 
XX,  i». 


Acetic  acid  (25°),  CH3COO'—  H'  

....    1,800,000 

Solution. 
I  .  * 

Carbonic  acid,  H'  —  HCO3'  

•?  040 

0.  174 

H-—  CO3" 

o  6f 

Hydrogen  sulphide,  H'—  HS'  

<C7O 

O.O7^ 

Boric  acid  H"  —  H2BO3' 

17 

O    Olt 

Hydrocyanic  acid  H'  —  CN'    

I  ? 

O.OII 

Phenol.  H-—  CJI.O'  .  . 

I  .  3 

0.0037 

In  the  case  of  the  addition  of  a  salt  with  an  ion  in 
common  to  an  organic  acid  (pp.  134-135  )  the  following 
will  be  seen  at  once  to  be  true.  If  the  degree  of  dissocia- 
tion of  a  salt  with  an  ion  in  common  with  an  acid  is  d, 
and  n  is  the  number  of  moles  per  liter  of  salt  present, 
then  the  equation  of  equilibrium  of  the  acid  will  become 


For  very  weak  acids  we  can  generalize  this  as  follows  : 
a,  the  degree  of  dissociation  of  the  acid,  is  very  small 
in  presence  of  the  salt,  so  that  a  in  comparison  to  i  and 
nd  may  be  neglected.  Since  d  for  salts  is  almost  inde- 
pendent of  the  dilution,  we  have 


KV 


affects  K  to  a  large  degree.  For  a  list  containing  a  very  large  number 
of  other  acids,  see  Zeit.  f.  phys.  Chem.,  3,  418-20,  1889,  or  Kohlrausch 
and  Holborn,  Leitvermogen  der  Elektrolyte,  pp.  176-193. 

*  Trans.  Chem.  Soc.,  77,  8,  1900. 

t  McCoy,  Am.  Chem.  Journ.,  29,  455,  1903. 


138  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

i.e.,  the  dissociation  0}  a  weak  acid  in  presence  of  one 
of  Us  salts  is  approximately  inversely  proportional  to 
the  concentration  oj  the  salt. 

The  case  of  the  partition  of  a  base  between  two  acids 
depends  to  a  certain  extent  upon  their  ionization  con- 
stants. This  partition  takes  place  when  there  is  not 
enough  base  present  to  saturate  both  acids.  The  final 
mixture  consists  of  water,  undissociated  salt,  and  the 
dissociated  and  undissociated  portions  of  the  acids. 
The  equilibrium  is  the  same  as  that  which  would  be 
attained  by  the  mixture  of  the  salt  of  the  one  acid  with 
the  other  acid.  The  affinity  for  each  acid  for  the  base 
will  depend  upon  the  percentage  of  ionized  H"  which 
it  possesses,  i.e.,  the  one  containing  the  larger  quantity 
will  unite  with  the  larger  amount  of  the  base. 

For  weak  acids  it  is  possible  to  formulate  a  general 
law  regarding  the  partition  and  the  ionization  constants. 
In  this  case  a  is  so  small  that  it  may  be  neglected  in 
i—  a;  hence  we  have  KV=a2,  i.e.,  a  =  \/KV.  And  for 
the  two  acids  at  the  same  dilution 


or     —=-_—. 
«     VK' 


The  coefficient  oj  partition  oj  two  acids,  then,  is  propor- 
tionate to  the  ratio  oj  their  degrees  oj  dissociation  at  the  given 

VK 

volume,  or  for  WEAK  acids  to     .  —  .-. 

VK' 

This  coefficient  is  independent  of  the  nature  of  the 
base,  and  depends  only  upon  the  two  acids. 

For  the  partition  of  an  acid  between  two  bases  the 
coefficient  depends  similarly  upon  the  two  bases,  and 
is  independent  of  the  nature  of  the  acid.  If  a  is  the 
degree  of  ionization  of  one  base,  and  a.'  that  of  the  other, 


EQUILIBRIUM  IN  ELECTROLYTES.  139 

we  shall  find  for  them,  when  weak,  just  as  for  acids,  that 

—  =  —  =,  and  the  same  generalization  holds  true. 
a'    V#' 

In  order  that  a  base  may  be  divided  equally  between 
two  acids  it  is  necessary  that  they  be  isohydric,  i.e.,  it 
is  necessary  that  Kv  =  K'v'.  An  example  of  such  iso- 
hydric solutions,  i.e.  two  which  contain  the  same  con- 
centration of  ionized  H",  is  acetic  acid  at  a  dilution  of 
8  liters,  and  hydrochloric  acid  at  one  of  667  liters.  These 
two  solutions  may  be  mixed  in  all  proportions  without 
any  change  in  the  ionization  resulting;  and  when  mixed 
in  equal  volumes,  if  treated  with  a  small  quantity  of  base, 
equal  amounts  of  chloride  and  acetate  will  be  formed. 

All  these  conclusions  for  organic  acids  hold  also  for  the 
organic  bases,  as  well  as  for  some  inorganic  ones.  The 
ionization  constants  for  these  when  binary  are  naturally 
found  from  the  relation 


where  M*  is  the  positive  and  OH'  the  negative  kind  of 
ionized  matter.  The  value  of  K  for  a  few  bases  is  given 
below  : 

Urea  (4Q°.2)  .....................  ^=0.0037X10-" 

Acetamide       (40°.2)  ......  ..  ..............  0.0033X  IQ-" 

Acetanilide      (4o°.2)  .....................  O.0044X  io~" 

Aniline  (25°)  ......................  5.4       Xio~10 

^-Toluidine    (4o°.2)  .....................  20.  7       X  io~10 

m-Nitraniline  (4o°.2)  .....................  4          X  io~12 

p.        "  (4Q°.2)  ....................  i  X  io~12 

o-        "  (40°.2)  .....................  o.oi     Xio~n 

Ammonium  hydrate,  NH^  —  OH'  (25°)  .....  23  X  io~ 

Methylamine          (25°).  ...  .....  5  X  io~ 

Dimethalamine      (25°)  ..................  0.074  X  io~ 

Trimethylamine     (25°)  ..................  o.oo74X  i 

Propylamine  (25°)  ..................  0-047  X  io~ 

Isopropylamine     (25°)  ..................  o.  053  X  io~ 

Isobutylamine       (25°)  ..................  0.034  X  io~ 


14°  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

The  equation  on  pages  134-135  has  also  been  used  to 
determine  the  concentration  of  ionized  matter  in  the  solu- 
tion added  to  a  substance  obeying  the  law  of  mass  action, 
and  with  very  good  results.  Thus,  starting  with  acetic 
acid,  knowing  the  concentration  of  ionized  H',  and 
adding  sodium  acetate,  we  can  find  the  concentration  of 
ionized  CH3COO'  in  the  salt.  In  this  case  the  new 
concentration  of  ionized  H'  can  be  determined  from  the 
speed  of  inversion  of  sugar  (p.  130);  and  by  solving  the 
equation  for  the  amount  of  CH3COO'  added,  we  can 
find  the  ionization  of  the  salt  solution  added. 

This  same  process  may  also  be  carried  out  with  other 
systems,  mentioned  below,  so  that  by  it  it  is  possible  to 
determine  the  concentration  of  any  one  kind  of  ionized 
matter.  As  mentioned  above,  the  ionization  has  been  found 
to  be  sensibly  the  same,  by  whatever  method  it  may  be  de- 
termined. 

Acids,  bases,  and  salts  which  are  ionized  to  a  con- 
siderable extent.  Empirical  dilution  laws. — The  appli- 
cation of  the  law  of  mass  action  (the  Ostwald  dilution 
law)  to  the  above  so-called  strong  electrolytes  does  not 
lead  to  a  constant  value  for  K  when  the  ionization  is 

it 
taken  from  the  conductivity  ratio  — .     As    in   general 

/*£» 

the  degree  of  electrolytic  dissociation  is  found  to  be  the 
same  by  all  of  the  possible  methods,  our  only  conclusion 
at  present  is  that  the  law  of  mass  action  cannot  be  applied 
to '  the  equilibrium  of  un-ionized  and  ionized  portions  in 
such  solutions  as  these.  Although  this  is  true  with  regard 
to  very  soluble  salts,  there  is  a  quantitative  relation, 
which  we  shall  develop  below,  holding  for  the  ionized 
portions  of  difficultly  soluble  substances,  when  the  un- 
ionized portion  is  retained  constant, 


EQUILIBRIUM  IN  ELECTROLYTES.  14 * 

The  only  known  case  of  a  dissociating  and  very  soluble 
salt  to  which  the  law  of  mass  action  may  be  applied,  i.e. 
the  only  exception  to  the  above  conclusion,  is  caesium 
nitrate,  when  a  is  determined  by  the  freezing-point 
method.*  And  this  is  not  true  for  a  as  determined  by 
the  other  methods.  In  other  words,  in  this  case,  the 
freezing-point  results  point  to  a  different  degree  of  ioni- 
zation  than  any  other  method.  The  results  for  caesium 

a2 

nitrate  as  determined  by  Biltz  show  that  K=  (  _    ^=0.34 

for  the  freezing-point  determinations  of  a,  while  the 
value  calculated  by  aid  of  conductivity  shows  no  trace  of 
constancy. 

Biltz  attributes  the  failure  of  the  law  of  mass  action, 
as  applied  to  strong  electrolytes,  to  a  hydration  of  the 
substance,  i.e.  to  a  chemical  reaction  between  the  solute 
and  the  solvent,  which  would  remove  active  solvent 
from  the  solution,  leaving  it  really  more  concentrated 
than  it  appears  to  be.  Just  why  the  solution  of  caesium 
nitrate  in  water  should  give  a  constant  value  for  K  when 
a  is  determined  by  the  freezing-point  method,  while 
when  a  is  found  from  the  conductivity  it  fails  to  do  so, 
is  unknown  and  thus  far  nothing  but  assumption  has 
been  possible.  It  is  to  be  remembered,  however,  that 
this  value  of  K,  although  giving,  when  solved  for  a, 
the  ionization  according  to  the  freezing-point  method, 
does  not  give  the  value  determined  by  other  methods, 
so  that  too  much  stress  is  not  to  be  laid  upon  it,  espe- 
cially in  face  of  the  fact  that  the  law  cannot  be  applied 
to  any  other  strong  electrolyte,  determine  a  by  any 
method  that  we  will.  It  would  certainly  seem  more 

*  Biltz,  Zeit.  f.  phys.  Chem.,  40,  218,  1902;  also  "Elements,"  p.  294. 


142   PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

probable  that  in  this  one  case  some  secondary  action 
occurs  in  the  freezing  which  is  absent  in  all  other  cases, 
so  that  here  the  freezing-point  leads  to  an  incorrect 
(i.e.  abnormal)  result.  This  conclusion,  of  course,  may 
not  be  true,  but  at  the  present  time,  in  the  absence  of 
any  indication  of  such  a  result  for  other  substances,  it 
is  decidedly  the  most  reasonable  and  logical  one.  In 
other  words,  then,  so  far  as  we  know  to-day,  the  con- 
ductivity leads  in  all  cases  to  the  true  value  for  a,  hence 
it  is  to  be  inferred,  failing  further  evidence,  that  it  is  also 
correct  here,  and  we  must  attribute  the  case  above  to 
some  secondary  action  not  yet  encountered  in  studying 
other  substances. 

Although  the  Ostwald  dilution  law  (i.e.,  the  law  of 
mass  action)  fails  utterly  to  hold  for  the  equilibrium 
between  the  ionized  and  un-ionized  portions  of  a 
strong  electrolyte,  certain  other  empirical  dilution  laws 
have  been  found  which  allow  us  to  find  the  respective 
amounts  at  any  dilution.  Thus  Rudolphi  *  found  a 
dilution  law  which  gives  a  constant  value  within  certain 
limits  for  such  solutions  as  do  not  follow  the  Ostwald 


a2 
dilution  law.     This  law  is  -         —7=-  =  constant,  where 

(i-a)VV 

the  value  of  the  constant  is  approximately  the  same  for 
analogous  substances,     van't  Hoff  f  altered  this  to  the 

a3 

form  ; r^  =  constant,  which  holds  even  better  than 

(i-a)2V 

Q3 
Rudolphi's.    Simplified,  this  relation  is  -^  —  constant,  i.e., 

Cs 

the  cube  of  the  concentration  of  ionized  matter  divided  by 
the  square  oj  the  un-ionized  portion  is  a  constant. 

*Zeit.  f.  phys.  Chem.,  17,  385,  1895. 
f  Ibid.,  1 8,  300,  1895. 


EQUILIBRIUM  IN  ELECTROLYTES.  143 


Writing   the   Ostwald   dilution   law   in   this   form  we 

C2 

obtain   (for  binary  electrolytes)  K=—,  while    this    em- 

cs 

pirical  relation  (in  either  jorm)  remains  the  same  for 
binary  or  ternary  substances;  in  other  words,  is  inde- 
pendent of  the  number  o)  moles  of  ionized  matter  formed 
from  one  mole  of  the  substance. 

Bancroft  *    proposes    a    dilution    law    of    the    form 

c~n 

const.  =  — ,  in  which  the  constant  and  n  are  functions  of 
£« 

the  nature  of  the  electrolyte.  Although  this  relationship 
is  as  yet  unknown,  Bancroft  suggests  that  we  may  some- 
time find  a  relation  of  the  form  n  =  2  —  /  (constant) — 
where  /  (constant)  varies  between  zero  and  a  value  ap- 
proximating one-half — which  will  reconcile  the  Ostwald 
dilution  law,  holding  only  for  organic  acids  and  bases, 
and  in  which  n  =  2,  with  this  form  in  which  n  varies 
in  value  from  1.36  to  1.5.  For  KC1  at  18°  we  have 

£.1.36 

the   relation    -     —  =  2.63,  which  holds  in  a  very  remark- 
£« 

able  way  between   the   volumes  0.3   and    10,000   liters. 
Noyes  *  has  determined  the  degree  of  ionization,  ^, 

/*oo 

for  the  chlorides  of  potassium  and  sodium  at  various 
temperatures  and  finds  a  constant  value  for  the  ratio  — j— ; 

that  is,  the  fraction  of  salt  un-ionized  is  directly  proportional 
to  the  cube  root  of  the  concentration,  or  the  concentration 
of  un-ionized  substance,  (i-a)c,  is  directly  proportional  to 
the  4/3  power  of  the  total  concentration,  c,  of  salt.  The 

*  Zeit.  f.  phys.  Chem.,  31,  188,  1899.     (In  English.) 
f  J.  Am.  Chem.  Soc.,  26,  168,  1904. 


144  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

degrees  of  ionization  of  potassium  and  sodium  chlorides 
were  found  to  be  nearly  identical  (the  extreme  variation 
being  2%)  at  all  temperatures  and  dilutions.  In  a  o.i 
molar  solution  the  dissociation  has  approximately  the 
following  values : 

18°  84%  281°  67% 

140°  79%  306°  60% 

218°  74% 

The  values  of  K'  =  — ^—  are  as  follows : 

18°  140°  218°  281°  306° 

Nad 0.366          0.448          0.573  0-745  0-877 

KC1 0.321          0.468          0.577  0.713  0.853 

Some  of  the  observed  facts  as  to  the  ionization  of 
strong  electrolytes,  as  summarized  by  Noyes,*  are  as 
follows : 

The  form  of  the  concentration  function  is  independent 
of  the  number  of  moles  of  ionized  matter  into  which  one 
mole  of  salt  dissociates.  Instead  of  being  proportional 
for  di-ionic,  tri-ionic,  and  tetra-ionic  to  the  square,  cube, 
or  fourth  power  of  the  concentration  of  the  ionized  matter, 
the  un-ionized  portion  is  approximately  proportional  to 
the  3/2  power  of  that  concentration,  whatever  may  be 
the  type  of  salt. 

The  conductivity  and  freezing-point  depression  of  a  mix- 
ture of  salts  having  one  kind  of  ionized  matter  in  common 
are  those  calculated  under  the  assumption  that  the  degree 
of  ionization  of  each  salt  is  that  which  it  would  have 
if  present  alone  at  such  an  equivalent  concentration  that 
the  concentration  of  either  of  the  kinds  of  ionized  matter 
were  equal  to  the  sum  of  the  equivalent  concentrations  of 
all  the  positive  or  negative  ionized  matter  present  in  the 
mixture.  Suppose  that  a  mixed  solution  is  o.  i  molar  with 
*  Technology  Quarterly,  17,  307,  1904. 


EQUILIBRIUM  IN  ELECTROLYTES.  MS 

respect  to  sodium  chloride  and  0.2  molar  with  respect  to 
sodium  sulphate,  and  that  it  is  0.18  molar  with  reference 
to  the  positive  or  negative  ionized  matter  of  these  salts. 
The  principle  then  requires  that  the  ionization  of  either 
of  these  salts  in  the  mixture  be  the  same  as  it  is  in  water 
alone  when  its  ionic  concentration  is  0.18  molar.  This 
has  been  proven  conclusively  for  many  mixtures. 

The  decrease  of  ionization  with  increasing  concentra- 
tion is  roughly  constant  in  the  case  of  different  salts  of 
the  same  type. 

The  un-ionized  fraction  of  any  definite  molal  concen- 
tration is  roughly  proportional  to  the  product  of  the 
valences  of  the  two  kinds  of  ionized  matter  in  the  case 
of  salts  of  different  types.* 

From  these  facts,  together  with  others,  Noyes  con- 
cludes that  the  form  of  union  represented  by  the  un- 
ionized portion  of  a  substance  differs  essentially  from 
ordinary  chemical  combination,  it  being  so  much  less 
intimate  that  the  kinds  of  ionized  matter  exhibit  their 
characteristic  properties,  in  so  far  as  these  are  not  de- 
pendent upon  their  existence  as  separate  aggregates.  In 
other  words,  the  law  of  mass  action  is  inapplicable  to 
the  relation  between  the  ionized  and  un-ionized  portions 
as  they  exist  in  strong  electrolytes,  and  hence  this  is  not 
to  be  regarded  as  a  simple  chemical  equilibrium,  for  which, 
as  we  know,  the  law  of  mass  action  always  appears  to 
hold  rigidly. 

The  heat  of  ionization. — By  aid  of  van't  Hoff 's  equation 
(p.  119),  it  is  possible  to  calculate  the  heat  of  ionization 
of  a  substance,  provided  we  know  its  degree  of  ioniza- 
tion at  two  different  temperatures.  This  is  true  not 

*  Other  generalizations  of  this  kind  from  the  standpoint  of  electrical 
conductivity  ^ill  be  found  in  Chapter  VII, 


J46  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

only    for    those    substances    that    follow    the    Ostwald 
dilution    law,    but,    according    to    Arrhenius,  *   for    all 
others  as  well.     For  binary  electrolytes  we  have,  then, 
.       K'     .      a'2(i-o:)     q/i       i\      , 
l°S.=^2->  =          -->  where  the  values  of 


K  are  for  the  same  dilution  and  represent  the  change  of 
ionization,  even  though  the  values  are  not  the  same  as 
other  dilutions,  and  the  values  V  cancel  and  need  not  be 
considered,  q  is  then  the  heat  liberated  when  a  mole  of 
substance  is  formed  in  solution  by  the  union  of  the  kinds 
of  ionized  matter  composing  it.  It  will  be  observed  here 
that  these  values  differ  from  those  calculated  from  the 
table  given  on  page  95,  for  these  refer  to  the  heat  of  forma- 
tion of  ionized  matter  from  substance  already  in  solu- 
tion, while  those  refer  to  the  compound  process  of  solu- 
tion and  ionization,  i.e.,  the  difference  in  energy  between 
the  ionized  state  and  the  solid  or  gaseous  state.  Some 
of  the  results  as  found  by  Arrhenius  are  given  below, 
the  unit  being  the  small  calorie. 

HEATS  OF  IONIZATION. 

Substance.  Temperature.  Calories. 


Propionic  acid  ..................  |  ^o  ^  ~  557 

Butyric  acid...  ..{35^  ~  935 

Phosphoric  acid 


Hydrofluoric  acid  ...............  33°  —3549 

Potassium  chloride  ..............  35°  —   362 

1  '          iodide  ...............  35°  —  916 

bromide  ..............  35°  —   425 

Sodium  chloride  ................  35°  —   454 

'  '        hydrate  .................  35°  —  1292 

"       acetate  .................  35°  —   391 

Hydrochloric  acid  ...............  35°  —  1080 

*  Zeit.  f.  phys.  Chem.,  4,  96,  1889,  and  9,  339,  1892. 


EQUILIBRIUM  IN  ELECTROLYTES.  14.7 

HEAT  NECESSARY  TO  COMPLETE  THE  IONIZATION,  (i  —  a)w  (i  mole  in 

200  moles  of  water). 

Substance.  Temperature.  Calories. 

Potassium  bromide 35°  —     58 

"          iodide 35°  —   132 

"         chloride '. 35°  -     56 

Sodium  hydrate 35°  —    180 

chloride 35°  -     81 

Hydrochloric  acid 35°  —    136 

Hydrofluoric  acid 33°  —  3304 

Phosphoric  acid 21°. 5  — 1682 

For  the  temperatures  of  35°  in  the  table,  2^  =  273  +  18, 
^273  +  52;  for  2i°.5,  r  =  273  +  i8,  r  =  273  +  25  in 

177 

the  log*"^  formula  (p.  119). 

From  data  such  as  the  above  it  is  possible  to  calculate 
the  heat  of  neutralization  of  an  acid  by  a  base.  The 
formula  for  this  (p.  92)  is 

q=x+w3(i  —a3)—w2(i  —a2)  —Wi(i  — «i), 

where  the  figures  i,  2,  and  3  refer  respectively  to  acid, 
base,  and  salt,  and  x  is  the  heat  of  formation  of  i  mole 
of  water  from  ionized  H*  and  ionized  OH',  i.e.,  13,700 
cal.  In  the  table  below  the  calculated  values  of  q  at 
two  temperatures  are  given,  together  with  the  observed 
values  at  one  of  the  temperatures. 

HEAT  OF  NEUTRALIZATION  OF  ACIDS  WITH  NaOH. 
(i  mole  of  acid  + 1  mole  of  NaOH 4-400  moles  of  H,O.) 
At  35°.  At  21.5. 

Calc.  Calc!  ObsT" 

HC1 12867  13447  13740 

HBr 12945  13525  13750 

HNO3 12970  13550  13680 

CH3COOH 13094  13263  13400 

C^COOH 13390  13598  13480 

CHC12COOH 14491  14930  14830 

H3PO4 14720  14959  14830 

HF 16184  16320  16270 


148  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

It  is  obvious  from  the  results  above  that  the  value  oj 
the  heat  0}  neutralization  oj  an  acid  by  a  base  cannot  be 
considered  as  indicative  of  the  strength  of  the  acid.  The 
two  latter  are  relatively  weak  acids  and  yet  they  give 
rise  to  the  greatest  amount  of  heat. 

This  formula  of  van't  HofFs,  as  was  mentioned  above 
can  only  be  used  to  calculate  the  heat  of  solution  of  sub- 
stances  which  are  completely  ionized  (or  practically  so) 
or  completely  un-ionized.  And,  naturally,  until  our 
knowledge  of  the  conditions  governing  equilibrium  in 
such  systems  is  considerably  broadened,  we  cannot  expect 
to  find  a  formula  that  will  hold. 

Solubility  or  ionic  product.  —  Although,  as  we  have 
seen,  the  law  of  mass  action  cannot  in  general  be  applied 
to  the  equilibrium  of  the  ionized  and  un-ionized  portions 
of  a  substance  in  solution  (except  to  organic  acid  and 
bases),  it  can  be  applied  with  considerable  accuracy  to 
a  very  large  number  of  saturated  solutions.  An  example 
of  such  an  equilibrium  is  a  saturated  solution  of  silver 
chloride,  which  is  found  to  be  practically  completely 
ionized  according  to  the  scheme 


AgCl=Ag' 
Applying  the  law  of  mass  action  to  this  we  obtain 


or        = 


(i-a)F' 


when  c  is  the  concentration  of  un-ionized  AgCl,  c\  that  of 
ionized  Ag",  and  c^,  that  of  ionized  Cl'.  Since  the  solu- 
tion is  saturated,  the  value  of  c  at  any  temperature  must 
remain  constant,  for  if  the  solution  were  unsaturated, 
solid  would  dissolve,  or  if  supersaturated,  solid  would  pre~ 


EQUILIBRIUM  IN  ELECTROLYTES.  H9 

cipitate.  In  a  saturated  solution  of  silver  chloride,  at 
any  one  temperature,  then,  we  have  the  relation 

Kc  =  constant  =  CiC2  ; 

i.e.,  in  a  saturated  solution  of  a  binary  electrolyte  (of  this 
kind)  the  product  oj  the  concentrations  oj  the  two  kinds  0} 
ionized  matter  must  remain  constant,  with  unchanged  tem- 
perature. 

Expressing  this  in  a  more  general  form,  we  have  for 
the  reaction 


nA 
in  a  saturated  solution,  the  relation 

ci"lc2H*  =  constant  =  s, 

where  5  has  been  called  by  Ostwald  the  solubility  product 
of  the  substance.  This  solubility  product  is  of  para- 
mount importance  in  analytical  chemistry,  for  a  precipi- 
tate (when  due  to  an  ionic  reaction,  and  most  of  them 
can  be  shown  to  be  due  to  this)  is  always  and  only  formed 
when  its  solubility  product  is  exceeded.  This,  of  course, 
presupposes  that  no  supersaturation  phenomenon  is  pos- 
sible ;  if  it  is,  then  the  so-called  metastable  limit  *  must 
first  be  exceeded. 

Just  as  we  found  a  decrease  in  the  dissociation  of  a  gas 
or  an  organic  acid,  by  the  addition  of  one  of  the  products 
of  dissociation  from  an  exterior  source,  so  here  the  addi- 
tion of  a  substance  with  a  kind  of  ionized  matter  in  common 
causes  the  formation  and  separation  in  the  solid  state  of 
the  un-ionized  substance.  In  other  words,  the  term  5  still 

*"  Elements,"  p.  128. 


ISO  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

retains  its  constant  value,  and  consequently  the  kinds  of 
ionized  matter  composing  the  substance  unite  to  form 
more  of  the  un-ionized  portion,  which,  since  the  solution 
is  already  saturated  with  it,  separates  out  as  solid.  This 
has  been  found  to  be  true  by  experiment,  but  only  true 
quantitatively  for  those  substances  which  are  difficultly  sol- 
uble* The  effect  may  be  observed  most  easily  by  dissolv- 
ing the  difficultly  soluble  substance  in  a  solution  of  the 
salt  with  ionized  matter  in  common;  but  it  can  also  be 
attained  by  adding  to  the  saturated  water  solution  of 
the  substance  a  strong  solution  of  the  salt,  when  a  pre- 
cipitation of  the  substance,  usually  in  the  crystalline 
state,  will  be  observed.  Thus  if  we  add  to  one  portion 
of  a  saturated  solution  of  silver  acetate  a  strong -solution 
of  sodium  acetate  containing  x  moles  of  ionized 
CH3COO',  and  the  same  amount  of  a  solution  of  silver 
nitrate  containing  x  moles  of  ionized  Ag*  to  the  liter 
to  another  equal  portion,  we  observe  an  equal  precipi- 
tation of  solid  silver  acetate  in  the  two  solutions. 

The  examples  below  will  serve  to  show  how  the  solu- 
bility product  of  a  substance  can  be  found,  and  how 
when  once  found  it  can  be  employed  to  foresee  the  solu- 
bility of  the  substance  in  a  solution  already  containing  a 
common  kind  of  ionized  matter. 


*Noyes  and  Abbott  (Zeit.  f.  phys.  Chem.,  16,  138,  1895)  have  found 
for  those  substances  which  are  largely  dissociated,  and  this  is  general, 
that  the  concentration  of  the  un-ionized  part  of  the  salt  has  always  the 
same  value  when  the  product  of  the  concentrations  of  the  kinds  of  ionized 
matter  it  produces  has  the  same  value,  whatever  may  be  the  values  of 
the  two  separate  factors  of  that  product.  In  other  words,  if  the  strong 
electrolyte  AD,  ionizing  into  A'  and  D',  has  a  concentration  of  AD 
in  a  saturated  solution  equal  to  y,  when  A'XD'  has  the  value  x,  it 
will  also  have  the  value  of  y  whenever  A '  X  D'  has  the  value  of  x,  whether 
it  be  produced  by  (A'  —  z)(D'+z}  or  (A' +z)(D'-z),  etc. 


EQUILIBRIUM  IN  ELECTROLYTES.  15* 

Silver  bromate  is  soluble  at  25°  to  the  extent  of  0.0081 
moles  per  liter.  Assuming  the  ionization  in  this  state  to 
be  practically  complete,  and  it  certainly  is  nearly  so,  the 
concentration  of  the  ionized  Ag*  and  ionized  BrO3'  will 
be  the  same,  and  equal  each  to  0.0081  mole  per  liter. 
The  solubility  product  at  this  temperature,  then,  will  be 

(0.0081  )  (0.0081  )  =  5AgBr0l. 

The  solubility  in  a  solution  of  silver  nitrate  containing 
o.i  mole  of  ionized  Ag*  (or  in  potassium  bromate  con- 
taining o.i  mole  of  ionized  BrO3')  can  now  be  found 
readily  by  aid  of  the  relation 

(0.0081  )2  =  (0.0081  +.1  -;y)(o.oo8i  -y\ 

and  is  equal  to  (0.0081  -y),  for  that  is  the  concentration 
of  ionized  Ag*  and  ionized  BrO3'  now  existing  in  the 
solution,  and  coming  from  the  salt;  the  amount  o.i 
of  one  being  due  to  the  other  salt,  and  y  being  the  AgBrO3 
remaining  undissolved  owing  to  the  presence  of  this 
o.i  mole  of  Ag-  or  BrO3'. 

This  is  true  for  all  binary  salts  when  they  can  be  as- 
sumed to  be  completely  ionized,  or  practically  so. 

Where  the  substance  dissociates  into  more  than  two 
kinds  of  ionized  matter,  and  can  be  assumed  to  be  com- 
pletely ionized,  the  relation  is  quite  similar.  Suppose 
the  salt  to  dissociate  according  to  the  scheme 


As  solubility  product  we  shall  have,  if  c  is  the  solubility 
of  the  completely  ionized  salt  MA&  c  X  (^c)3  =  sMA3>  or 
for  we  must  nave  three  times  the  num- 


IS2  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

her  of  moles  per  liter  of  A'  as  we  have  of  M'"  accord- 
ing to  the  chemical  equation,  i.e.,  c  =  CM-~  and  3^  =  c^.  In 
case  of  solution  in  the  presence  of  o.  i  mole  of  one  of  the 
kinds  of  ionized  matter,  we  have,  then,  either 


(%•••+  o.  i  - 
or  C—XC    +0.1  - 


from  which  it  is  apparent  that  the  effect  of  equal  ad- 
dition is  not  the  same  for  the  two  kinds  of  ionized  matter, 
i.e.,  that  x  and  y,  the  decreases  in  the  solubility,  are  not 
equal. 

In  the  case  the  substance  is  not  completely  ionized,  the 
solubility  product  is  not  so  directly  related  to  the  solu- 
bility of  the  substance  as  in  the  above  cases,  i.e.,  to  the 
square  in  one  case  and  twenty-seven  times  the  fourth 
power  in  the  other.  Consider  the  case  of  uric  acid,  which 
at  25°  is  soluble  to  0.0001506  mole  per  liter,  and  is 
ionized  in  that  condition  to  9.5%  into  H*  and  the  ionized 
negative  radical  which  we  shall  designate  as  V.  The 
solubility  product  here  is  naturally 

(0.0001506X0.095)  (0.0001506X0.095) 

=  SHU  =  #HU(O.OOOI  506  Xo.905). 


The  solubility  of  uric  acid  in  a  molar  solution  of  hydro- 
chloric acid,  for  which  a  =  0.78  (i.e.  H*=o.78,  Cl'  =  o.78), 
is  to  be  found  in  the  following  way: 

(o.oooi  506  Xo.o95  +  0.78—  x)  (o.oooi  506  Xo.095  —  x) 

=  (0.0001506  Xo.095)2, 

where  (0.0001506X0.095—  #)  represents  the  present  con- 
centration of  H*   and  U'  from  the  uric   acid,   and  its 


EQUILIBRIUM  IN  ELECTROLYTES.  153 

total  solubility  in  the  hydrochloric  acid  solution  is 
(0.0001506X0.905) +  (0.0001506X0.095—3;),  i.e.,  is  equal 
to  the  sum  of  that  which  is  un-ionized  and  that  which  is 
ionized. 

In  general,  just  as  for  organic  acids,  an  infinite  excess 
of  one  of  the  ions  will  cause  the  ionization  of  the  sub- 
stance to  become  zero.  It  is  to  be  observed  here,  however, 
that  this  excess  will  only  cause  the  solubility  to  become  zero 
in  the  case  that  the  ionization  is  complete.  In  the  case  of 
uric  acid,  an  infinite  amount  of  H*  or  U'  at  best  can  only 
reduce  the  solubility  by  9.5%,  the  remaining  90.5% 
being  un-ionized  and  not  affected  at  that  temperature  by 
any  addition  of  substance  which  does  not  react  chem- 
ically with  it. 

That  a  substance  is  always  decreased  in  solubility  by 
the  addition  of  a  substance  with  ionized  matter  in  common 
is  not  true,  as  the  well-known  behavior  of  silver  cyanide  in 
potassium  cyanide  will  show.  In  all  such  cases,  howeve^ 
the  equilibrium  which  has  previously  existed  is  altered 
in  some  way,  so  that  the  relations  are  not  the  same- 
These  cases  are  usually  characterized  by  the  formation  of 
a  complex  kind-  of  ionized  matter  the  product  of  which 
is  exceeded.  The  removal  of  the  kinds  of  ionized  matter 
necessary  to  form  this  complex  kind  disturbs  the  equi- 
librium of  the  difficultly  soluble  salt;  the  un-ionized  por- 
tion then  ionizes  further,  and  its  loss  is  replaced  by  the 
solid  phase.  This  process  continues,  dissolving  new  salt, 
until  equilibrium  is  attained,  i.e.,  until  the  solubility  prod- 
uct, whatever  it  may  be,  is  just  satisfied,  when  solution 
ceases.  The  ionization  of  silver  potassium  cyanide  takes 
place  almost  completely  according  to  the  scheme 

KAgCN2=K'+AgCN2', 


154  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

but  it  has  been  found  by  Morgan  *  that  in  a  0.05  molar 
solution  we  have  ionized  Ag*  to  the  extent  of  3.5Xio~n 
and  ionized  CN7  to  2.76Xio~3  moles  per  liter. 

Knowing  the  concentration  of  the  ionized  metal,  for 
example  (which  can  be  determined  by  methods  given 
in  the  next  chapter),  in  the  complex  salt  solution  and  in 
a  water  solution  of  the  difficultly  soluble  salt,  we  can  fore- 
see the  behavior  of  that  salt  when  in  a  solution  of  a  salt 
which  might  dissolve  it  to  form  a  complex  solution  of 
that  strength.  In  general,  i.e.,  when  the  concentration 
oj  ionized  metal  in  a  water  solution  of  salt  is  greater  than 
that  of  a  water  solution  of  a  complex  salt,  the  simple  salt 
will  dissolve  in  the  solution  which  will  produce  the  com- 
plex salt  in  this  concentration.  If  the  concentration  oj 
ionized  metal  is  smaller  the  solid  will  not  dissolve  to  any 
greater  extent  than  it  does  in  pure  water,  jor  the  ionic 
product  oj  the  ionized  complex  cannot  be  exceeded. 

By  this  law  it  is  possible  to  find  the  relative  solubility 
of  salts  of  the  same  metal  in  water.  Thus  silver  sulphide 
is  the  only  silver  salt  which  will  not  dissolve  in  potassium 
cyanide  solutions;  in  other  words,  is  the  most  insoluble 
salt  of  silver,  and  contains  less  ionized  Ag*  in  a  saturated 
solution  than  exists  even  in  a  solution  of  silver  potassium 
cyanide,  such  as  that  given  above. 

It  is  not  only  for  substances  in  solution  that  we  find 
this  constancy  of  the  product  of  the  concentrations  of 
the  kinds  of  ionized  matter,  for  it  also  exists  in  our 
usual  solvent,  water,  where  the  ionized  portion  is  so  small 
that  the  un-ionized  portion  may  be  considered  as  con- 
stant, i.e.  i -a  does  not  differ  appreciably  from  i. 
Expressing  the  concentrations  of  ionized  H*  and  ion- 
ized OH'  in  a  liter  of  water  by  Ci  and  c2,  and  the 

*Zeit.  f.  phys.  Chem.,  17,  513-535,  1895. 


EQUILIBRIUM  IN  ELECTROLYTES.  155 

un-ionized    portion,    which  is    practically    i     liter,    i.e. 

1000 

— —  =55.5  moles,  by  c,  we  have 

Io 

5  5 .  5^H2o  =  SHZO  =  constant. 


The  values  of  c\  =  c2  =  ionized  H"   (  =  ionized  OH')  in 
water  at  various  temperatures  is  as  follows: 

Temp.         Moles  per  Liter.  Temp.  Moles  per  Liter 


0°  0.35  34°  1.47 

10°  0.56  50°  2.48 

18°  0.80  85°.  5  6.20 

25°  1.09  1.00°  8.50 


The  ionic  products  (we  can  hardly  call  them  solubility 
products),  then,  are  as  follows: 

*o°=(o.35XIO~7)2>  534  =  (1.47X10 -7)2, 

SW°  =  (0.56X10  -7)2,  s50=  (2.48X10  -7)2, 

Sls°  =  (0.80  X  10  -7)2,  s85,5°  =  (6. 2  X  10  -7)2, 

*25°  =  (1 .09  X  10  -7)2,  S10(J°  =  (8. 5  X  10  -7)2, 

where  the  value  of  s,  in  each  case,  is  equal  to  55.5  times 

a2 
the  value  of  K=(   _    .„,  V  being  0.0018  liter,  i.e.,  the 

volume  occupied  by  the  formula  weight,   18  grams,  of 
water. 

Knowing  the  solubility  products  of  two  substances 
with  a  kind  of  ionized'matter  in  common,  it  is  possible 
to  find  how  much  of  each  will  dissolve  when  a  mixture 
of  them  is  exposed  to  the  action  of  a  solvent;  and  this, 


I56  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

of  course,  may  be  expanded  to  three  or  more  substances 
together. 

Assume  we  have  the  two  completely  ionized,  diffi- 
cultly soluble  salts  MA  and  MAi,  with  the  ionized  mat- 
ter M'  in  common,  and  that  they  are  dissolved  simul- 
taneously in  water.  Call  the  amount  of  MA,  which 
dissolves,  x,  and  the  amount  of  MAi  y.  In  the  solution, 
then,  we  must  have  x+y  moles  of  ionized  M',  x  of  ionized 
A'  and  y  of  ionized  A\r\  and,  if  s  is  the  solubility  product 
of  M  A  and  s\  that  of  MAi,  the  relations  must  be 


so  that  by  solving  the  simultaneous  equations  we  can 
find  x  and  y. 

An  example  of  this  is  given  by  dissolving  thallium 
chloride  and  sulphocyanate  together.  The  solubilities 
in  water,  each  for  itself,  are  TlCl  =  o.oi6i  and  T1SCN 
=0.0149.  Assuming  complete  ionization,  the  solubility 
products  are  respectively  (o.oi6i)2  and  (0.0149)2,  an<^ 
if  x  represents  the  amount  of  chloride  and  y  that  of 
sulphocyanate  dissolving  from  the  mixture,  we  have 
Tl',  x  =  Cl',  and  y  =  SCN',  and 


x(x+y)  =  (o.oi6i)2, 
y(x+y)  =  (0.0149)2, 

from  which  we  find  x  =  0.0118  and  ;y=o.oioi,  while 
the  values  #  =  0.0119  and  ;y  =  0.0107  are  found  by  ex- 
periment. 

It  will  be  observed  that  in  the  above  examples,  except 
the  last,  we  have  tacitly  assumed  that  the  dissociation 


EQUILIBRIUM  IN  ELECTROLYTES.  157 

of  the  added  salt,  with  a  kind  of  ionized  matter  in  com- 
mon, is  not  influenced  by  the  same  kind  of  ionized  matter 
from  the  difficultly  soluble  salt.  As  a  rule  this  is  true, 
for  the  substances  are  so  insoluble  that  their  effect  is  infini- 
tesimal; in  the  last  example,  this  effect  has  been  allowed 
for,  however,  and  will  show  the  method  of  treating 
such  cases. 

In  general,  then,  we  can  conclude  for  difficultly  soluble 
salts  (and  jor  ionized  complexes)  that  they  are  precipitated 
(formed)  when  the  product  of  the  concentrations  of  the 
ionized  substances  composing  them  exceeds  the  solubility 
(ionic)  product.  Although  this  law  holds  in  general  for 
difficultly  soluble  salts,  isolated  cases  are  to  be  found 
where  the  un-ionized  portion  does  not  remain  rigidly 
constant,  after  the  addition  of  a  substance  giving  the 
same  kind  of  ionized  matter  in  common;  and,  to  a 
smaller  extent,  a  slight  variation  is  sometimes  observed 
in  the  solubility  product.  Since  these  cases  are  very 
few,  and  are  usually  observed  for  the  more  soluble  salts, 
it  would  seem  probable  that  they  are  due  to  secondary 
reactions  not  yet  recognized,  or  to  others  not  properly 
accounted  for.* 

Hydrolytic  dissociation,  or  hydrolysis.  Hydrolysis  is 
the  process  taking  place  in  a  water  solution  of  a  saltt 
which  causes  the  solution  to  appear  alkaline  or  acid, 
or  results  in  a  neutral  equilibrium  according  to  the  scheme 

MA+H20  =  MOH+HA. 

If  the  acid  formed  is  insoluble  or  un-ionized,  the  base 
being  ionized,  the  reaction  will  be  alkaline  (action  of 
ionized  OH').  When  the  base  is  insoluble  or  un-ionized, 

*  See  foot-note,  p.  150. 


158  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

and  the  acid  ionized,  the  reaction  is  acid  (action  of 
ionized  H')-  And,  finally,  if  both  acid  and  base  are 
insoluble  or  un-ionized,  the  salt  is  completely  transformed 
into  base  and  acid,  and,  as  there  will  remain  no  excess 
of  either  OH'  or  H',  the  reaction  will  be  neutral.  In 
other  words,  then,  hydrolysis  is  the  name  by  which  we 
designate  the  process  resulting  from  the  removal  of  either 
H'  or  OH'  (or  both)  from  the  water  by  the  ionized  A' 
or  the  ionized  M'  of  the  salt,  to  form  un-ionized  or  in- 
soluble substances ;  in  short,  since  this  removal  causes  the 
further  ionization  of  the  water,  hydrolysis  is  the  chemical 
process  observed  to  take  place  between  a  salt  and  water. 

Examples  of  this  process  are  most  common.  For 
instance,  all  mercury,  copper,  zinc,  etc.,  salts  are  acid, 
for  un-ionized  basic  substances  (for  which  the  ionic 
products  are  exceeded)  are  formed  by  the  reaction  with 
water,  leaving  free,  ionized  acid;  and -potassium  cyanide 
is  alkaline,  due  to  the  formation  of  un-ionized  hydro- 
cyanic acid,  and  ionized  potassium  hydrate. 

Since  we  know  the  conditions  under  which  insoluble 
or  un-ionized  substances  will  form,  i.e.  by  the  exceeding 
of  their  solubility  products  or  analogous  values,  it  is 
possible  to  find  the  conditions  necessary  to  produce  a 
hydrolytic  dissociation,  and  to  calculate  the  extent  of 
this  when  it  does  take  place,  i.e.,  to  find  the  equilibrium 
which  is  finally  attained  in  the  solution. 

We  recognize  at  once  that  if  the  product  of  the  con- 
centrations of  the  ionized  M'  and  the  ionized  OH'  is 
larger  than  that  which  can  exist  in  pure  water,  un-ionized 
substance  must  form.  By  this  formation,  however,  the 
equilibrium  of  H*  and  OH'  will  be  disturbed,  and  a  further 
ionization  of  water  must  take  place,  until  at  length  the 
ionic  product  is  just  attained.  If  the  H'  and  A'  at  this 


EQUILIBRIUM  IN  ELECTROLYTES.  159 

point  do  not  unite  to  form  un-ionized  acid,  the  further 
ionization  of  water  will  be  unlike  what  it  would  be  in 
the  absence  of  this  excess  of  ionized  H*,  for,  since 
'  must  at  tne  same  time  be  equal  to  sH2O,  we 


can  only  have  —  ^  moles  per  liter  of  ionized  OH'  present, 

when  CH-  is  the  total  concentration  of  ionized  H"  at  that 
time. 

The  process  due  to  the  formation  of  un-ionized  or 
insoluble  acid,  when  no  un-ionized  base  is  formed,  or 
forms  but  slightly,  is  exactly  analogous  to  the  above.  In 
both  cases  water  is  decomposed,  owing  to  the  removal 
of  one  of  its  two  kinds  of  ionized  matter,  and  the  further 
ionization  of  water  and  the  formation  of  the  insoluble  or 
un-ionized  base  or  aeid  continues  until  the  equations 
representing  equilibrium  are  fulfilled. 

For  the  sake  of  simplicity  we  shall  first  consider  sepa- 
rately the  cases  that  the  reaction  is  caused  by  the  base 
or  by  the  acid. 

Case  I.  The  process  is  due  primarily  to  the  formation  0} 
base.  Here  it  is  obvious  that 

^M-X^OH'>(^MOHCMOH)     or     >%OH» 

where  the  terms  c  refer  to  the  ionic  concentrations.  Call- 
ing c  the  original  concentration  of  salt,  and  ds  the  ioni- 
zation of  the  salt,  the  concentration  of  ionized  OH'  in 
water  at  25°  being  LopXio"7  moles  per  liter,  we  shall 
have 

dsc  X  1.09  X  10  -7  >  (^MOH^MOH)     or     >  JMOH. 


After  equilibrium   has  been  established,  i.e.  when  the 
degree  of  hydrolytic  dissociation  is  a,  </A  being  the  dis- 


160  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 
sociation  of  the  acid  formed,  we  must  have 

total  OH' = 


total  H* 
and 

(orig.  M'-loss  of  M")  — ^^.  =  #MOHXMOH  formed, 


where,  if  the  base  has  a  solubility  product,  it  is  to  be  used 
in  place  of  the  terms  on  the  right,  and  the  value  sH2o 
varies  with  the  temperature,  having  the  value  (1.09  X  10  ~7)2 
at  25°. 

Case  II.     The  process  is  due  primarily  to  the  formation 
of  acid.    Here 

cA'  X  CH-  >  (#HA^HA)     or     >  <>  HA- 


Just  as  above,  since  Cn-  =  i.ogXio~7  at  25°,  we  shall 
have,  when  c  is  the  concentration  of  salt,  and  d$  its 
ionization, 


or 


And  if  dB  is  the  ionization  of  the  base,  c  the  original 
concentration  of  salt,  and  a  its  hydrolytic  dissociation, 
then 

1  IT*—         ^HeO 

and 
(orig.  A' -loss  of  Ao(tot^QH,)  =  #HAXHA  formed, 


EQUILIBRIUM  IN  ELECTROLYTES.  161 

or 


And  here  again  the  solubility  product  may  be  used  on 
the  right  if  the  acid  is  difficultly  soluble. 

The  formulas  above,  in  both  cases,  may  also  be  written 
in  another  form,  which,  although  it  does  not  illustrate 
so  well  the  principles  involved,  is  more  useful  in  many 
ways,  for  it  enables  us  to  obtain  a  constant  of  hydrolytic 
dissociation,  from  which  the  value  of  a  at  any  other  dilu- 
tion and  that  temperature.  From  page  160,  by  trans- 
formation, we  obtain 

Cds(l  -a)sH2Q  =  #MOH«2^A> 

or 

<*2       _  %2o      d_ 
' 


(i-a)V'ds         '*• 
and  from  the  above  equation,  in  the  same  way, 


_ 

(i-a)V'  ds          d-     *HA'     (i-a)F 
In  dilute  solutions  where  ds,  dA,  and  d-Q  may  be  regarded 
as  unity,  these  equations  are  simplified  to  the  form 


__ 


"nyd.       /T       ,v  M7        V 

(i—a)V     AHA 

We  have  the  following  law,  then,  governing  hydrolysis. 
The  expression  for  the  hydrolytic  dissociation  of  a  salt 

a2 
in  water, r^,  is  equal,  when  due  to  the  formation  of 

base  (acid),  to  the  ionic  product  of  water  multiplied  by 
the  degree  of  ionization  of  the  salt  divided  by  the  ionization 
constant  of  the  base  (acid),  multiplied  by  the  degree  of 
ionization  of  the  acid  (base). 


162  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

This  law  has  recently  been  much  used  to  determine 
from  the  experimentally  observed  hydrolysis  of  the  salt 
the  ionization  constant  of  the  acid  or  base  formed. 

The  constant  of  hydrolytic  dissociation,  in  such  a  case, 
can  also  be  defined  (see  pp.  107  and  135),  when  d$,  d&, 
or  d&  are  equal  to  i,  as  one- half  the  concentration,  in 
moles  per  liter,  at  which  the  salt  is  50%  hydrolyzed. 

And  if  a  is  small  enough  to  be  neglected  in  the  term 

a2 

i— a,      _    \y  =  K  is  also  reduced  (p.  138)  to 

KV=a2; 

in  other  words,  for  the  same  substance,  the  hydrolytic 
dissociation,  when  small,  is  proportional  to  the  square  root 

of  the  dilution  of  the  salt,  i.e.,  aoc\/V    or    >J— ,  where 

c,  the  reciprocal  of  V,  is  the  original  concentration  of  the 
salt  dissolved. 

Knowing  the  constant  for  hydrolytic  dissociation  it  is 
also  possible  to  calculate  the  degree  of  hydrolysis  at  any 

\~       K2v2    KV 
dilution  by  the  formula  a  =  \JKV+ X .      The 

following  examples  will  serve  to  show  the  use  which 
may  be  made  of  the  above  relations. 

What  is  the  ionic  product  for  water  at  25°?  A  o.i 
molar  solution  of  sodium  acetate  is  0.008%  hydrolyzed, 
the  sodium  acetate  to  be  considered  as  completely  ionized, 
as  is  also  the  sodium  hydrate  formed,  and  the  ionization 
constant  of  acetic  acid  is  0.000018. 

Here 

CH3COOH  =  OH'  =  0,00008  X  o.  i  =  0.000008, 


EQUILIBRIUM  IN  ELECTROLYTES.  163 

and  since 


0.000018  Xo.oooooS 
£H-=—         =1.44X10  * 


and 


What  is  the  hydrolysis  of  a  o.  i  molar  solution  of  potas- 
sium cyanide  (assuming  ds=  i)  ?  K  for  HCN  =  13  X  10  ~10 
and  sH2o  =  (i-°9Xio-7)2  at  25°. 


a2          (1.09  X  10  ~7)2 
hy(L~(i-a)F~    13X10-!°    ' 

from  which,  when  d%  =  i  and  F=io,  a  =0.967%. 

a2 
In  the  table  below  are  given  the  values  of  -  -  r-~  for 

various  equilibria,  in  which  but  i  mole  of  water  reacts 
with  the  substance. 

HYDROLYSIS  OF  HYDROCHLORIDES  AT  25°. 


Aniline  ...............     2.7  2.25Xio~5  5.3Xio~10 

o-Toluidine  ...........     7.0  1.62X10-*  7.3  X  io~n 

m-        "        ..........     3-6  4-ioXio-5  2.9  Xio-10 

p-        "        ..........     1.8  1.05X10-*  1.13X10-° 

o-Nitroaniline  .........  98.  6  2.1  5-6X  io~  15 

m-         "  ........   26.6  3.01X10-'  4.0  Xio~12 

p-          "          ........   79.6  9.58X10-*  i.24Xio-13 

Aminoazobenzene  .....   18.1  1.25X10-*  9.5   Xio~10 

Urea  ....................  0.781  i.5Xio-u 

Thus  far  we  have  only  considered  that  one  mole  of 

water  reacts  with  the  salt;  in  other  words,  we  have  only 


164  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

employed  salts  containing  monovalent  elements.  In 
case  the  reaction  involves  more  than  one  mole  of  water 
the  treatment  is  the  same  as  in  any  other  application  of 
the  law  of  mass  action.  It  must  be  said,  however,  that 
such  cases,  so  far  as  we  know  at  present,  are  not  at  all 
common,  the  salt  often  reacting  with  but  one  mole  of  water 
to  form  a  basic  salt  which  still  retains  some  of  the  original 
element.  There  is  one  reaction,  however,  which  gives  a 
good  constant  assuming  two  moles  of  water  to  react,  and 
we  shall  consider  it  to  show  how  such  relations  are  to  be 
treated.  The  reaction  is 

A1C18  +  2H20  =  A1(OH)2C1  +  2HC1, 

which  has  been  investigated  by  Kullgren.*  We  have, 
tnen,  where  c  is  the  molar  concentration  of  AlCla  in  solu- 
tion, and  a  is  the  fraction  of  it  hydrolyzed,  d$  being  the 
ionization  of  AlCls,  and  d&  that  of  the  HC1,  of  which 
the  total  amount  20.0  is  formed, 

/  s        \2 
2         = 


It  is  impossible  to  use  this  formula  in  calculations,  how- 
ever, for  as  yet  we  know  nothing  of  KA1(OH)2ci-  But 
by  using  the  equation  in  the  other  form  (p.  161),  we 
obtain 

4q;3      dj         ^H2o         „ 
' 


and  from  this  we  can  calculate  the  ionization  constant 
of  A1(OH)2C1,  when  a  is  known,  or  dispense  entirely 

*  Om    metallsalters    hydrolys,     p.     108.     Dissertation.     Stockholm. 
1904. 


EQUILIBRIUM  IN  ELECTROLYTES.  165 

with  it,  i.e.,  using  the  Khyd.  so  determined  for  the  cal- 
culation of  other  values.  It  will  be  observed  here  that  a, 
instead  of  being  proportional  to  VV  as  it  is  for  the  reac- 
tion with  i  mole  of  water,  is  proportional  to  VF*.  The 
following  results  will  show  how  well  this  equilibrium 
follows  the  above  law,  and  how  it  is  possible  to  find  the 
ionization  constant  by  aid  of  the  hydrolytic  dissociation, 
knowing  the  ionic  product  for  water  at  that  temperature. 

HYDROLYSIS  OF  A1C13  AT  100°  C. 
A1C13+3H20=A1(OH)2C14-2H' 


96         0.1488         0.966         0.76         420X10-'         516X10-' 

384         0.3629         0.977         0.85         509X10-'         571X10-' 

1536         0.7142         i  0.91         541X10-'         594X10-' 

Average,     ^"hyd.  =  5  60  X  i  o~' 

The  concentrations  of  base  in  the  three  cases  are  0.00155 
0.000945,  and  0.000465,  respectively,  the  acid  concen- 
trations being  twice  these  values.  The  average  value  of 
•^hyd.  in  tne  last  column  may  be  used  to  determine 

Hz°  T'    We  obtain  ^ 

ds 


this  way  the  value    ^Ai(OH)2ci  =  2.33Xio-1»,  where  the 
ionization,  presumably,  gives  A1C1"  and  2  OH'. 

The  formation  of  a  substance  containing  OH  as  well 
as  the  original  negative  element  is  very  common.  In 
the  case  of  the  chloride  of  bismuth  mentioned  above  the 
substance  separating  out  by  hydrolysis  is  not  the  pure 
hydrated  oxide,  but  an  oxychloride,  although  apparently 
this  is  not  true  in  the  case  of  the  hydrolysis  of  ferric 
chloride.  In  this  case  the  reaction  is  not 

FeCl3  +  3H20  -  Fe(OH)3  +  3H'  +  3C1', 


1 66  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 
but,  according  to  Goodwin,*  must  rather  be 

FeCl3 +H20  =  FeOH"  +H*  +3C1', 

where  the  FeOH"  is  colloidal. 

In  certain  other  cases  it  has  been  found  that  hydrolytic 
dissociation  takes  place  in  stages,  i.e.,  first  i  mole  of 
water  reacts,  then  another,  etc.  It  is  quite  certain,  how- 
ever, that  this  does  not  occur  at  the  dilutions  above  of 
AlCla,  for  if  it  did,  the  formula  used  would  not  give  a 
constant  value,  hence  in  this  one  case  between  these 
limits  of  dilution  2  moles  of  water  react  with  i  of  salt. 

T1(NO3)3,  according  to  Spencer  and  Abegg,f  on  the 
other  hand,  seems  to  react  directly  with  3  moles  of  water, 
the  T1(OH)3  having  a  solubility  equal  to  io~13-58  moles 
per  liter,  i.e.,  s  =  io~52-896,  but  as  yet  this  is  the  only 
case  known. 

Naturally,  any  method  for  determining  the  concen-' 
tration  of  ionized  H*  or  ionized  OH',  or  the  undisso- 
ciated  substance  formed,  will  enable  us  to  find  the  amount 
of  hydrolytic  dissociation. 

One  method,  which  can  be  used  for  salts  of  weak  acids 
with  strong  bases,  or  salts  of  weak  bases  with  strong 
acids,  has  been  suggested  by  Farmer,  {  and  this,  owing 
to  the  importance  of  the  principle  involved,  is  briefly 
considered  below.  The  method  is  based  upon  the 
coefficient  of  distribution  of  a  substance  between  water 
and  another  solvent,  benzene  (pp.  74-75).  Thus,  hydroxy- 
azobenzene  has  a  coefficient  of  distribution  between 
water  and  benzene  equal  to  539,  i.e.,  benzene  always 

*  Zeit.  f.  phys.  Chem.,  21,  i,  1896,  and  Phys.  Rev.,  u,  193,  1900.  • 

f  Zeit.  f.  anorg.  Chem.,  44,  397,  1905. 

J  Trans.  Chem.  Soc.,  79,  £63,  1901,  and  ibid.,  85,  1713,  1904. 


EQUILIBRIUM  M  ELECTROLYTES.  167 

takes  up  539  times  as  much  hydroxyazobenzene  as  the 
water,  when  the  two  solvents  are  present  in  equal  volumes. 
If  the  two  solvents  are  present  in  unequal  quantity,  say 
i  liter  of  water  to  q  liters  of  benzene,  the  hydroxyazo- 
benzene in  the  water  will  be  distributed  between  them 
in  the  ratio  i :  539*7. 

By  shaking  a  water  solution  of  the  barium  salt  of 
hydroxyazobenzene  with  benzene,  then,  the  free  hydroxy- 
azobenzene, if  it  be  formed  by  hydrolysis,  will  be  partially 
extracted  by  the  benzene.  Finding  the  amount  of  this 

present  in  the  benzene  solution,  multiplying  it  by  , 

we  find  what  is  left  in  the  aqueous  solution.  The  sum 
of  these  two  quantities,  then,  is  the  concentration  of  free 
hydroxyazobenzene  which  has  been  formed  as  the  result 
of  hydrolysis,  and,  knowing  the  amount  of  salt  initially 
present,  the  degree  of  hydrolytic  dissociation  is  easily 
calculated.  By  this  method,  for  numerous  dilutions  of 

a2 
the  barium  salt,  the  formula  J£==     _    .„  was  found  to 

give  a  constant  value  for  K,  which  at  25°  is  equal  to 
24.3X10-7. 

This  method  has  also  been  applied  to  the  hydrolysis  of 
the  hydrochlorides  of  weak  bases,  as  aniline,  etc.  (where 
the  coefficient  of  distribution  of  the  free  base  is  deter- 
mined), with  very  satisfactory  results.  The  values  in  the 
table  on  page  163  were  found  in  this  manner. 

In  all  such  determinations  constancy  of  temperature  is 
of  paramount  importance,  for  hydrolytic  dissociation,  as 
will  have  been  observed  from  the  foregoing,  is  largely  in- 
fluenced by  the  temperature.  This  is  due  not  only  to  the 
increased  ionization  of  water  (p.  155)  with  the  temper- 
ature, but  also  to  the  decrease  in  the  ionization  constants 


168  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

of  acids  and  bases.  Thus  for  acetic  acid  Kl8°=^  18.3 X 
10-6,  tfioo.  =  n.4Xio-«,  KI56*  =  5.6X10 -6,  and  K2l8*  = 
1.9X10-6,  while  for  ammonium  hydrate  Kl8°=i'j.iX 
10-6,  £loo.  =  14X10-*,  and  #I56o=  6.6X10 -6. 

Ionic  equilibria.  —  Knowing  the  solubility  or  ionic 
products,  and  the  ionization  constants  of  the  constitu- 
ents, it  is  often  possible  to  gain  an  idea  of  the  mechanism 
of  the  reaction  and  the  equilibrium  that  will  be  produced; 
or,  on  the  other  hand,  to  calculate  some  of  these  factors 
when  the  composition  of  the  system  at  equilibrium  is 
known.  Thus  it  is  possible  to  calculate  the  ionization 
constant  of  an  acid  (or  a  base)  from  its  increased  solu- 
bility in  a  base  (or  an  acid)  with  a  known  ionization 
constant.*  In  the  same  manner,  also  by  aid  of  the  law  of 
mass  action,  it  can  be  proven  that  magnesium  hydrate  is 
not  precipitated  by  ammonia  in  the  presence  of  ammonium 
chloride,  not  because  of  the  formation  of  a  double  salt  of 
magnesium  and  ammonium,  but  simply  because  the  de- 
crease in  the  amount  of  ionized  OH'  from  NH±OH,  by 
the  presence  of  an  excess  of  ionized  NH±  from  the  NH^Cl, 
is  so  great  that  the  solubility  product  of  Mg(OH)2  cannot 
be  exceeded.^  And  this  conclusion  has  been  confirmed  by 
Treadwell,|  who  studied  the  freezing-points  of  solutions 
of  MgCl2  and  NH4C1  separately,  and  then  when  present 
together,  and  showed  conclusively  that  no  such  compound 
exists  in  solution. 

Findlay  §  has  investigated  the  reversible  reaction 

*  See  "  Elements,"  pp.  329-334;  Lb'wenherz,  Zeit.  f.  phys.  Chem.,  15, 
385,  1898. 

t  "  Elements,"  pp.  339-342;  Loven,  Zeit.  f.  anorg.  Chem.,  37,  327, 
1896;  Muhs,  Dissertation,  Breslau,  1904. 

J  "Elements,"  pp.  342-345;  Zeit.  f.  anorg.  Chem.,  37,  327,  1903. 

§  "Elements,"  pp.  345-346;  Zeit.  f.  phys.  Chem.,  34,  409,  1900. 


EQUILIBRIUM  IN  ELECTROLYTES.  169 

solid  PbSO4+ dissolved  2NaI<z»  solid  PbI2 

+ dissolved  Na2SO4; 
i.e.,  expressed  in  ionic  form, 

solid  PbSO4  +  2l'<=±  solid  PbI2+SO4". 

Applying  the  law  of  mass  action  to  this  he  found  that 

4 

=  constant, 

Cso4" 

where  at  25°  C.  the  value  of  the  constant  lies  between 
0.25  and  0.3.  The  outcome  of  the  investigation  may  be 
summed  up  as  follows:  From  a  mixed  solution  of  so- 
dium iodide  and  sodium  sulphate,  by  the  addition  oj  a 
soluble  lead  salt,  pure  lead  iodide  (the  more  soluble)  can 
be  precipitated  if  the  ratio  of  the  square  oj  the  concentra- 
tion of  ionized  iodine  to  the  concentration  of  the  ionized 
sulphate  radical  (i.e.,  SO 4")  is  greater  than  the  equilib- 
rium constant.  When  the  ratio  becomes  equal  to  this 
constant,  both  lead  iodide^  and  sulphate  are  precipitated 

together,  the  ratio  -    —  remaining  constant.     And  all  this 

CS04" 

is  true  for  the  sulphate  (the  less  soluble)  when  the  ratio  is 
smaller  than  the  constant. 

It  will  be  observed  from  these  examples  that  by  aid 
of  the  law  of  mass  action,  even  though  it  fails  to  hold 
for  strong  electrolytes,  we  can  foresee  and  regulate  many, 
if  not  most,  of  the  reactions  with  which  we  come  in  con- 
tact. Many  other  examples  could  be  cited  here  to 
illustrate  the  methods  of  application,  but  the  few  above 
will  suffice  to  bring  out  the  general  principles  and  enable 
the  reader  to  follow  work  of  this  sort. 

Naturally,  the  solubility  product,  and  the  other  con- 
ceptions developed  above,  are  of  great  importance  in 


17°  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

analytical  chemistry.  For  an  account  of  these  the 
reader  must  be  referred  *  elsewhere,  however,  as  it  is, 
only  the  principles,  and  not  so  much  the  application, 
that  can  be  included  in  this  small  volume. 

The  color  of  solutions. — The  color  of  a  solution  de- 
pends apparently  upon  the  condition  of  the  solute  in 
the  solvent.  If  a  substance  is  not  at  all  ionized,  or 
but  slightly  so,  any  color  it  may  possess  must  be  attrib- 
uted to  the  un-ionized  substance.  In  case  the  ionization 
is  practically  complete  the  color  of  the  solution  will  be 
the  result  of  the  mixture  of  the  colors  of  the  kinds  of 
ionized  matter  present;  or  if  only  one  kind  is  colored,  that 
color  will  be  the  color  of  the  liquid.  When  partly  disso- 
ciated, then,  the  color  of  a  solution  will  be  the  result  of 
the  mixture  of  the  colors  of  the  ionized  and  the  un-ionized 
portions;  or  if  only  one  of  these  is  colored,  that  color  will 
be  the  color  of  the  liquid.  The  un-ionized  portion  in 
cases,  however,  may  also  show  the  color  of  the  ionized 
portion,  f 

There  is  always  a  chance  of  error  here  if  the  color 
of  the  solid  is  assumed  to  be  its  color  in  solution.  The 
color  of  a  crystal,  for  example,  is  very  often  different 
from  that  of  the  substance  in  the  form  of  powder,  and, 
further,  it  may  be  possible  that  a  dissociation  takes  place 
in  the  water  of  crystallization.  In  this  latter  case,  of 
course,  the  solid  would  exhibit  the  same  color  as  the 
colored  ion.  The  only  correct  way  to  find  the  color 
of  the  undissociated  portion  in  solution  is  to  use  a  solvent 
in  which  the  substance  is  not  dissociated  to  any  extent, 
then  the  color  can  be  directly  observed.  This  is  not 

*  See  "Elements,"  pp.  349-363;   Ostwald's  Scientific  Foundations  of 
Analytical  Chemistry;    or  Bottger's  Qualitative  Analyse, 
f  Noyes,  Technology  Quarterly,  17,  306,  1904. 


EQUILIBRIUM  IN  ELECTROLYTES.  171 

difficult  to  carry  out,  for  all  solvents  have  a  different 
ionizing  power,  and  either  alcohol,  ether,  benzene,  chloro- 
form, or  acetone  will  be  found  to  serve  the  purpose. 

The  ionized  matter  produced  from  most  acids  is  color- 
less, consequently  all  salts  of  a  metal  in  very  dilute  solu- 
tions will  have  the  same  color,  i.e.,  the  color  of  the  ionized 
metal.  In  more  concentrated  solutions  this  is  not  true, 
for  many  un-ionized  substances  are  colored  and,  as  they 
are  now  present  to  a  greater  amount,  the  color  of  the 
solution  is  the  result  of  the  mixture  of  the  colors  of  these 
and  those  of  the  kinds  of  ionized  matter.  An  example  of 
this  is  given  by  solutions  of  cuprous  chloride,  where  the 
color  of  the  un-ionized  portion  is  yellow.  But  the  ionized 
copper  is  blue,  hence  the  color  of  a  solution  of  cuprous 
chloride  may  be  either  yellow,  green,  or  blue,  according 
as  it  is  undissociated,  or  ionized  to  a  lesser  or  greater 
degree.  All  copper  solutions  when  very  dilute,  how- 
ever, provided  the  negatively  charged  ionized  matter  is 
colorless,  show  the  same  blue  color. 

The  formation  of  an  ionized  complex  can  be  followed 
very  closely  when  it  is  produced  by  a  colored  and  a  color- 
less kind  of  ionized  matter.  Thus  if  a  KCN  solution 
is  added  to  a  colored  copper  solution  the  color  instan- 
taneously disappears,  due  to  the  formation  of  the  ionized 
complex  CuCN4".  The  formation  of  an  ionized  com- 
plex ion  can  be  proven,  of  course,  in  this  way,  but  its 
nature  can  only  be  shown  by  migration  experiments,  as 
will  be  shown  later.  (See  Chapter  VII.) 


CHAPTER  VII. 

ELECTROCHEMISTRY. 

THE  MIGRATION  OF  IONIZED   MATTER. 

Faraday's  law.  Electrical  units. — Before  discussing  the 
changes  produced  in  a  solution  by  the  passage  of  the 
electric  current,  we  must  first  consider  the  funda- 
mentals upon  which  the  measurement  of  such  a  change 
depends. 

The  general  guiding  principle  of  electrochemistry  is 
Faraday's  law,  which  regulates  the  relationship  between 
the  quantity  of  electricity  flowing  and  the  amount  of 
substance  it  deposits.  This  law  may  be  summarized  by 
the  two  following  statements: 

1.  The  amount  of  any  substance  deposited  by  the  current 
is  proportional  to  the  quantity  of  electricity  flowing  through 
the  electrolyte. 

2.  The  amounts  of  different  substances  deposited  by  the 
same  quantity  of  electricity  are  proportional  to  their  chem- 
ical equivalent  weights. 

In  order  to  make  use  of  this  law,  however,  it  is  neces- 
sary to  recall  the  definitions  of  the  various  electrical 
units  employed  in  describing  an  electric  current. 

The  resistance  offered  by  a  body  to  the  electric  cur- 
rent is  expressed  in  ohms,  I  ohm  being  the  resistance  at 
o°  C.  of  a  column  of  mercury  106.3  cms-  l°ng  with  a 

172 


ELECT  ROCHEM1STR  Y.  1  7  3 

cross-section  of  i  square  mm.  This  is  equal  to  io9 
absolute  units. 

The  unit  of  electromotive  force  is  the  volt,  i.e.,  io8  abso- 
lute units.  A  Daniell  cell  has  an  electromotive  force 
of  i.io  volts. 

The  ampere  is  the  unit  of  current  strength;  i  ampere 
will  separate  0.001118  gram  of  silver  from  a  solution  in 
i  second,  and  is  equal  to  lo"1  absolute  units. 

These  three  quantities  are  related,  according  to  Ohm's 
law,  in  such  a  way  that  in  any  one  circuit  we  always 
find 

electromotive  force 

current  strength  =  —       —  r-r  —  . 

resistance 

The  actual  quantity  of  current,  i.e.  amperes  per  second, 
is  expressed  in  coulombs:  i  gram  of  ionized  H*  carries 
with  it  96,540  coulombs. 

As  the  intensity  factors  of  electrical  energy,  then,  we 
have  the  voltage,  while  the  capacity  factor  is  expressed 
in  coulombs,  i.e. 


where  E  is  the  electrical  work,  the  unit  of  which  is  the 
watt-second,  equal  to  an  ampere  at  i  volt  in  i  second. 
This  is  equal  to  io7  absolute  units. 

The  heat  equivalent  of  electrical  energy,  since  the  unit 
of  the  latter  is  equal  to  volt  X  coulomb  or  io7  absolute 
units,  is 


i.e.,  i  watt-second  =0.2394  calories. 


174  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

To  separate  i  gram  of  hydrogen,  then,  or  the  equiva- 
lent weight  in  grams  of  any  other  element,  we  require 
the  work 

TT  X  96540  X  watt-seconds  =  96540  X o.  239471  =  23 1  ion  cals. 

The  migration  of  ionized  matter. — The  chemical  effect 
of  the  passage  of  an  electric  current  through  an  electrolyte 
can  be  divided  into  two  distinct  portions,  viz.,  the  con- 
duction through  the  electrolyte  and  the  separation  of 
substance  at  the  electrodes.  It  is  not  necessary  that 
the  ionized  matter,  which  serves  for  the  conduction  of 
the  current  through  the  liquid,  be  separated  at  the 
electrodes,  for  secondary  reactions  may  take  place  there, 
causing  other  substances  to  appear  as  the  result  of  the 
electrolysis.  It  is  to  be  remembered  here,  however,  that 
even  in  such  a  case  Faraday's  law  still  holds,  and  the 
substances  separated  are  chemically  equivalent  in  amount 
to  those  which  would  have  been  separated  in  case  the 
secondary  action  had  been  avoided. 

Although  the  two  different  effects  are  observed  to- 
gether in  practice,  we  shall  consider  them  separately, 
taking  up  the  question  of  conduction  here  and  putting 
off  that  of  separation  to  a  later  period. 

The  conduction  through  the  liquid  depends  upon  what 
we  have  designated  thus  far  as  ionized  matter,  and  varies 
according  to  the  mobility  of  this,  which,  in  turn,  is  de- 
pendent upon  the  specific  nature  of  the  matter,  its  con- 
centration, the  temperature,  and  the  nature  of  the  solvent. 

After  the  electrolysis  of  a  solution,  excluding,  or  allow- 
ing for,  any  secondary  reaction,  it  is  found  experimentally 
that  the  concentrations  around  the  anode  and  cathode 
are  not  always  identical,  as  they  were  initially.  In 
few  cases  they  are,  it  is  true,  but  in  these  it  can  be 


ELECTROCHEMISTRY. 


'75 


shown  (as  will  be  done  later)  that  the  mobility,  i.e. 
velocity  through  the  liquid  at  any  voltage,  is  the  same 
for  both  the  anion,  which  goes  to  the  anode,  as  it  is  for 
the  cathion,  which  goes  to  the  cathode.  In  all  other 
cases  the  mobility,  or  speed  of  migration  through  the 
liquid,  is  different  for  the  two  kinds  of  ionized  matter 
of  which  the  substance  is  composed. 

And  further  than  this,  the  analysis  of  the  anode  and 
cathode  liquids  after  electrolysis,  excluding  secondary 
reactions,  leads  to  an  expression  for  the  relative  mobilities, 
i.e.,  the  migration  ratio  of  the  two  kinds  of  ionized  mat- 
ter. Indeed,  when  the  original  solution  is  colored,  this 
difference  in  concentration  can  be  observed  qualitatively 
by  the  eye.  The  reasons  for  this  change,  together 
with  the  principle  upon  which  its  quantitative  calcula- 
tion is  based,  will  be  made  clear  by  the  following  con- 
siderations : 

Assume  the  vessel  in  the  figure  below  to  be  divided  into 
three  portions,  AC,  CD,  and  DB,  and  filled  with  a  solution 
containing  30  gram  equivalents  of  HC1.  We  have,  then, 


10  gram  equivalents  in  each  division.  If  96,540  cou- 
lombs of  electricity  are  passed  through  the  cell  from 
A  to  B,  i  gram  equivalent  of  ionized  H*  and  i  gram  equiv- 
alent of  ionized  Cl'  will  lose  their  charges  and  be  separated 
upon  the  electrodes  B  and  A,  if  secondary  action  is  ex- 
cluded. These  gases  we  assume  to  be  removed  as  they 
are  formed.  These  96,540  coulombs  passing  as  they  do 
tro  ugh  the  whole  solution  have  a  certain  effect  upon  the 


176  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

equilibrium  of  the  ionized  substances.  First  we  will  im- 
agine the  ionized  H'  and  ionized  Cl'  to  move  with  the 
same  velocity  and  then  with  differing  velocities,  and  find 
the  relation  between  the  change  in  concentration  and 
the  relative  mobilities. 

It  is  to  be  remembered  here  that  two  oppositely 
electrified  bodies  composing  a  system  will  transport  a 
current  equal  to  the  sum  of  the  charges  carried  by  the 
two  bodies  in  the  opposite  direction,  for  a  negative 
charge  going  in  one  direction  is  equivalent  to  an  equal 
and  opposite  charge  going  in  the  contrary  direction.  In 
other  words,  all  of  the  current  may  be  transported  by 
the  positive  material,  or  a  portion  may  be  carried  by 
each  in  opposite  directions,  and  in  all  cases  the  total 
current  is  the  sum  of  those  currents  going  in  the  opposite 
directions. 

I.  If  the  velocity  for  each  kind  of  ionized  matter  is  the 
same,  then,  1/2  gram  equivalent  of  ionized  Cl',  charged 
with  48,270  coulombs,  will  migrate  from  BD  through  DC 
to  CA ;  and  x/2  gram  equivalent  of  ionized  H',  with  the 
same  amount  of  electricity,  will  go  from  AC  through 
CD  to  DB.  Altogether,  then,  i  gram  equivalent,  charged 
with  96, 540  coulombs,  will  have  passed  through  the  section 
CD.  Since  i  gram  equivalent  of  ionized  H'  has  been 
removed  as  gas  by  decomposition  from  BD,  and  J/2  gram 
equivalent  has  migrated  to  it,  we  have  left  9^  gram 
equivalents  of  ionized  H*  and  9^  gram  equivalents  of 
ionized  Cl',  since  but  1/2  gram  equivalent  of  this  has 
migrated  from  it.  Consequently  we  have  in  BD,  9 J  gram 
equivalents  of  HC1.  In  AC  we  have  the  same  number, 
since  i  gram  equivalent  of  ionized  Cl'  has  disappeared, 
1/2  gram  equivalent  has  migrated  to  it  and  l/2  gram 
equivalent  of  ionized  H*  has  migrated  from  it.  In  the 


ELECTROCHEMISTR  Y.  1 7  7 

section  DC  the  concentration  is  unaltered,  i.e.,  just  as 
much  ionized  matter  has  left  it  as  has  been  carried  to  it. 

The  concentration  at  the  anode  is  the  same  as  that  at 
the  cathode,  then,  after  the  electrolysis  0}  a  solution  com- 
posed of  two  kinds  oj  ionized  matter  with  the  same  mobility. 

II.  Assume  the  velocity  of  the  ionized  H*  to  be  five 
times  that  of  ionized  Cl'. 

In  this  case,  after  i  gram  equivalent  of  H  and  i  gram 
equivalent  of  Cl  have  separated  in  the  gaseous  state, 
the  whole  system  will  have  suffered  a  change.  5/6  of  a 
gram  equivalent  of  ionized  H',  charged  with  4(96,540) 
coulombs,  will  migrate  from  A  C  through  DC  to  BD,  and  Ve 
of  a  gram  equivalent  of  ionized  Cl',  with  J  (96, 540)  cou- 
lombs, will  go  from  BD  through  CD  to  AC.  Altogether, 
as  before,  i  gram  equivalent  of  ionized  matter  will  go 
through  the  section  CD,  carrying  with  it  96,540  coulombs 
of  electricity. 

The  original  composition  of  the  solution  in  CD  is  again 
unchanged.  In  BD  we  have  lost  i  gram  equivalent 
of  ionized  H*  in  the  form  of  gas,  and  gained  5/6  of  a  gram 
equivalent  by  the  migration;  consequently  we  have  9! 
gram  equivalents  of  ionized  H*  left.  1/6  of  a  gram 
equivalent  of  ionized  Cl'  has  migrated  from  it,  so  that  in 
DC  we  have  9!  gram  equivalents  of  HC1. 

In  A  C  we  have  lost  5/6  of  a  gram  equivalent  of  ionized 
H'  and  i  gram  equivalent  of  ionized  Cl'  as  gas,  but  have 
gained  Ve  of  a  gram  equivalent  of  ionized  Cl'  by  the 
migration;  consequently  we  have  9^  gram  equivalents 
of  HC1  left. 

From  these  two  examples  the  following  law  may  be 
deduced:  The  loss  on  the  cathode  (BD)  is  related  to  that 
on1  the  anode  (AC)  as  the  mobility  oj  the  anion  matter 
(Cl')  is  to  that  oj  the  cathion  matter  (H'). 


178  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

In  this  way  Hittorf  determined  the  relative  mobilities 
or  migration  ratios  of  the  various  kinds  of  ionized  matter. 

The  practical  determination  of  the  relative  mobilities 
is  merely  a  matter  of  analysis.  The  apparatus  which  is 
used  for  this  purpose  is  a  decomposition-cell,  so  arranged 
that  no  metal  can  drop  from  one  electrode  to  the  other; 
or  a  U  tube  may  serve  the  purpose  so  long  as  the  two 
portions  of  liquid  may  be  removed  and  analyzed  sepa- 
rately. The  apparatus  is  filled  with  solution  and  the 
current  passed  through  for  a  certain  length  of  time,  the 
electrodes  being  of  the  metal  which  is  contained  in  the 
salt  or  inert,  except  in  cases  where  certain  secondary 
actions  are  to  be  avoided.  After  a  certain  time  either 
the  anode  or  cathode  portion  is  withdrawn  and  analyzed. 
This  analysis  will  give  us  the  loss  of  metal  on  the  one 
electrode,  from  which  that  on  the  other  may  be  calcu- 
lated. 

If  n  is  the  fraction  of  the  cathion  which  has  migrated 
from  the  anode  to  the  cathode  when  one  gram  equiva- 
lent has  been  separated,  then  i  —  n  is  that  fraction  of 
the  anion  which  has  gone  to  the  anode.  These  two 
quantities,  n  and  i  —  n,  are  called  the  migration  ratios 
of  the  cathion  and  anion.  We  have  then 

n         loss  at  anode       U 


i—n    loss  at  cathode      Ua 

where  Ue  is  the  mobility  of  the  cathion  and  Ua  that  of 
the  anion. 

An  example  will  make  the  determination  of  this  clear: 
Hittorf  electrolyzed  a  solution  of  AgNO3  until  1.2591 
grams  of  Ag  were  separated.  The  volume  of  liquid  at 
the  cathode  before  the  experiment  gave  17.46249  grams 


ELECTROCHEMISTR  Y.  1 7  9 

of  AgCl,  and  after  it  but  16.6796  grams,  i.e.,  a  loss  of 
0.7828  gram  of  AgCl  or  of  0.5893  gram  of  Ag. 

If  no  Ag  had  come  to  the  cathode  by  migration,  the 
solution  would  have  lost  1.2591  grams  of  Ag;  it  lost, 
however,  only  0.5893  gram;  hence  1.2591—0.5893  = 
0.6698  gram  Ag  has  come  to  it  by  the  migration.  If 
just  as  much  of  the  Ag  had  come  by  migration  as  had 
been  separated,  the  migration  ratio  of  the  ionized  Ag 
would  have  been  i,  i.e.,  the  ionized  NO3'  would  not  have 
migrated.  Only  0.6698  gram  of  Ag  has  migrated,  how- 
ever; hence  the  migration  ratio  for  the  ionized  Ag'  in 
AgNOa  can  be  found  from  the  proportion 

1.2591  :o.6698:  :i  1^=0.532; 
the  migration  ratio  of  the  ionized  NO3',  then,  is 
1—0.532  =  0.468. 

These  values  are  not  the  same  for  all  dilutions,  al- 
though in  general  the  variation  is  but  slight. 

A  table  containing  a  large  number  of  results  from 
experiments  of  this  sort  is  given  by  Kohlrausch  and  Hoi- 
born,  Leitvermogen  der  Elektrolyte,  a  few  of  which  will 
be  found  below. 

HITTORF'S  MIGRATION  RATIOS  FOR  TONIZED  MATTER. 
Solutions  i/io  equivalent  normal. 


Substance.  i— «. 

i/2K2S04 0.60 

1/2  CuSO4 0.64 

I/2H2S04 0.21 

i/2K2C03 0.37 

1/2  Na?CO, 0.48 

i/2Li2CO» 0.59 

KOH 0.74 

NaOH 0.84 

KC1 0.507 

Nad 0.63 

LiCl 0.70 


Substance.  i  —  ». 

HN4C1 0.508 

1/2  BaCLj 0.61 

i^CaCL, 0.68 

1/2  MgCl2 0.68 

HC1 0.21 

KNO3 0.50 

NaNO3 0.61 

AgNOs 0.526 

1/2  Ca(NO3)2 0.61 

KC1O3 0.46 


180  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

Naturally,  it  is  also  possible  here  to  use  inert  electrodes; 
and  in  many  cases,  where  a  secondary  reaction  is  to  be 
avoided,  it  is  necessary  to  employ  as  anode  a  metal 
which  differs  from  that  contained  in  the  salt. 

THE   CONDUCTIVITY   OF   ELECTROLYTES. 

The  specific,  molar,*  and  equivalent  conductivities. — 

The  conductivity  of  a  solution  may  be  determined  for 
the  same  volume  of  solution  (as  for  i  centimeter  cube,  the 
specific  conductivity)  independent  of  the  weight  of  sub- 
stance dissolved ;  or  for  the  volume  containing  i  formula 
weight  (according  to  the  generally  accepted  formula); 
or,  finally,  for  the  volume  containing  i  equivalent  formula 
weight  (i.e.,  the  weight  equivalent  chemically  to  i  gram 
of  hydrogen). 

The  unit  of  specific  conductivity  is  the  conductivity 
which  a  centimeter  cube  possesses  when  its  resistance  is 
i  ohm.  The  best  conducting  aqueous  solutions  of  the 
strong  acids  possess  this  conductivity  at  about  40°  C. 
We  shall  designate  specific  conductivities  by  K. 

This  specific  conductivity  is  the  unit  which  is  most 
employed  in  physics,  but,  since  the  conductivity  of  a 
solution  depends  almost  exclusively  upon  the  amount  of 
substance  it  contains,  it  is  far  more  convenient  to  apply 
chemical  conceptions  to  the  physical  fact,  and  express 
our  results  in  molar  or  equivalent  terms. 

The  equivalent  (molar)  conductivity  of  a  substance  is 
the  conductivity  0}  the  solution  which  contains  i  equiva- 
lent (i  mole)  of  substance,  the  electrodes  being  separated 
by  i  cm.,  and  large  enough  to  contain  between  them  the 

*  From  here  on  we  shall  employ  the  word  molar  conductivity  in  the 
sense  in  which  molecular  conductivity  was  used  above  (see  pages  77-79)- 


ELECTROCHEMISTRY.  181 

entire  solution.  This  value  can  be  found  by  dividing  K 
by  the  number  of  equivalents  (moles)  per  cubic  centi- 
meter, or  by  multiplying  K  by  the  number  of  cubic  centi- 
meters in  which  i  equivalent  (i  mole)  is  dissolved. 
When  V  is  the  volume  containing  i  equivalent  (or  i 
mole)  in  liters,  then,  KXioooXVe=A  and  «XioooXFm 
=  ft,  where  equivalent  conductivity  is  designated  by  A 
and  molar  conductivity  by  /JL. 

As  to  the  actual  measurement  of  the  specific  conduc- 
tivity, from  which  the  other  two  values  may  be  calculated, 
we  need  only  note  here  that  it  is  similar  to  the  ordinary- 
determination  of  resistance,  except  that  an  alternating 
current  is  used  in  place  of  the  direct,  and  the  galvanom- 
eter is  replaced  by  a  telephone  receiver.  Naturally,  the 
alternating  current  is  essential  here  to  prevent  actual 
decomposition,  which  would  produce  polarization  and 
cause  the  concentration  of  the  solution  to  decrease. 

Although  it  is  the  specific  conductivity  that  is  meas- 
ured, it  is  not  necessary  to  possess  a  set  of  electrodes 
exactly  i  cm.  in  cross-section  and  separated  by  exactly 
i  cm.,  nor  is  it  even  necessary  to  know  the  dimensions 
of  the  electrodes,  for  certain  conductivities  have  been 
accurately  measured  with  such  electrodes,  and  by  deter- 
mining the  value  of  one  of  these  solutions  in  any  form  of 
electrode  vessel  it  is  possible  to  find  a  constant  factor 
which  will  transform  results  obtained  with  it  into  specific 
conductivities.  Thus  for  a  0.02  molar  solution  of  KC1, 
Kohlrausch  found  the  values  «I8°  =  0.002397,  Al&0=  119.85, 
K25o  =  0.002768,  A25°  =  138.54. 

Ionic  conductivities. — Since  (by  definition,  p.  85)  the 
two  kinds  of  ionized  matter  are  the  carriers  of  electricity 
in  solution,  the  equivalent  conductivity  at  any  dilution 
divided  by  that  at  infinite  dilution,  i.e.  when  the  substance 


iS2  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

is  present  only  in  the  ionized  form,  gives  us  the  definition 
of  the^  degree  of  ionization.    We  have  then 


This  conductivity  at  infinite  dilution  means  simply  that 
the  equivalent  conductivity  is  not  altered  by  further 
dilution.  This  maximum  value  for  the  equivalent  con- 
ductivity Kohlrausch  found  for  a  binary  electrolyte  to  be 
equal  to  the  sum  of  two  single  values,  one  of  which  refers 
to  the  anion  matter  and  the  other  to  the  cathion  matter. 
This  law  of  the  independent  migration  of  ionized  matter 
shows  that  conductivity  is  an  additive  property.  The 
truth  of  the  law  is  shown  by  the  results  in  the  table  below  : 

MOLAR  CONDUCTIVITIES  AT  INFINITE  DILUTION. 

Ag 

109 

103 
83 

The  differences  of  two  corresponding  sets  of  numbers 
in  the  vertical  rows,  and  of  any  two  in  the  horizontal  ones, 
are  nearly  equal,  which  can  only  occur  when  the  result 
is  composed  of  two  single  and  independent  values. 

One  kind  of  ionized  matter,  then,  always  carries  the 
same  amount  of  electricity  with  its  own  velocity,  inde- 
pendent of  the  nature  of  the  ionized  matter  present 
with  it. 

The  equivalent  conductivity  at  infinite  dilution  is  con- 
sequently 


where  lc  and  la  are  the  equivalent  conductivities  of  the 


Cl 

K 

127 

Na 

IO3 

Li 
or 

NH4 
122 

H 
2C  7 

N03  
OH 

.  ..    118 
.    .   228 

98 
2OI 

35° 

CIO 

1  1  C 

C.KLO.,  . 

0/L 

77 

ELECTROCHEMISTRY.  183 

kinds  of  ionized  matter  produced  by  the  substance,  the 
solution  being  at  infinite  dilution. 
We  have  then  (p.  178) 

la 


in  other  words,     lc  =  nAx,     la  =  (i—ri)A^. 

Thus  the  molar  conductivity  at  infinite  dilution  ^w 
(equal  here  to  the  equivalent  conductivity  A^)  of  sodium 
chloride  is  no,  while  #  =  0.38  and  1—^=0.62;  hence 


i.e.,  I  mole  of  ionized  No,'  possesses  a  conductivity  of  41.8 
when  between  electrodes  I  cm.  apart  and  large  enough  to 
contain  between  them  the  total  volume  of  solution  in  which 
the  ionized  sodium  exists;  and  i  mole  of  ionized  Cl'  under 
the  same  conditions  has  a  conductivity  equal  to  68.2. 

In  all  solutions  in  the  same  solvent,  at  the  same  tem- 
perature, these  values  remain  constant,  so  that  it  is 
possible  for  us  to  calculate  what  the  conductivity  at  in- 
finite dilution  would  be  for  any  substance.-  This  is  of 
great  use  in  experimental  work,  for  it  is  not  always 
possible  to  actually  reach  this  limiting  value  with  any 
degree  of  accuracy. 

A  table  of  such  results,  then,  enables  us  to  find  the 
limiting  value  of  the  conductivity  at  infinite  dilution, 
and  not  only  this,  for 


i.e.,  if  we  know  the  fraction  of  a  mole  of  each  form  of  ionized 
matter  present  (by  any  other  method)  we  can  find  the 
equivalent  conductivity  of  the  solution  at  any  dilution  by 


184  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

multiplying  the  sum  of  the   ionic  conductivities    by  the 
degree  of  ionization. 

Since  most  neutral  salts  are  very  largely  ionized,  the 
value  of  the  equivalent  conductivity  can  be  readily  de- 
termined experimentally.  By  subtracting  the  value  for 
the  ionized  metal  from  this  result  it  is  possible,  then,  to 
find  the  ionic  conductivity  of  the  acid  radical  which  is 
present  as  negatively  charged  ionized  matter.  In  the 
table  below  are  given  a  few  ionic  conductivities,  from 
which  various  values  may  be  calculated. 


IONIC 
Li' 

CONDUCTIVITIES  AT  i 
/         Temp.  Coeff. 
•?•?  AA       o  026^ 

8°  AND  INFINITE 
IZn"  . 

DlLU' 

/ 

45  -6 
46.0 

56.3 
6i-5 
68.7 
70 
46.2 
64.7 
47-7 
53-4 
46.7 

35 
31 
27.6 

25-7 
24-3 

[TON.* 
Temp.  Coeff. 
0.0251 
0.0256 
0.0238 
0.0243 
0.0227 
0.0270 

Na"  

.    43."^       0.0244 

*Mg"  
ABa" 

K'  
Rb-  

.  .    64.67       0.0217 
.  .   67.  6        0.0214 

*Pb"  

Cs'  
NH'4  

.     68.2            0.0212 
.     64.4            O.O222 

iS04"  

*co,"  

Br03'  

CIO  ' 

xr  

Aff- 

.66            0.0215 

CA    O2         O   O22Q 

F'         .    . 

.   46  64       o  0238 

io/  

MnO4'  

CHcy  

C2H302'  
C3H60'  
C4H702'  
C5H902'  
CeHn02  



Cl'  

.   65.44       0.0216 

Br' 

.     67.63          O.02I5 

66  40      o  02  i  3 

I' 

SCN'  

CIO  ' 

.     56.63         0.02II 
re    Q3         O   O2I^ 



10,'  
NO,' 

.    33.87         0.0234 

61  78      o  0205 

H'  

.318 

OH'.. 

.174 

In  addition  to  the  above  ionic  conductivities,  and  the 
relative  mobilities,  it  is  also  possible  to  find  the  absolute 
mobility,  i.e.,  the  velocity  with  which  ionized  matter  of 


*  See  Kohlrausch,  Berl.  Akademieber.  26,  581,  1902,  and  Sitzungsber. 
d.  Akad.  d.  Wiss.  zu  Berlin,  26,  572,  1902. 


ELECTROCHEMISTRY.  185 

various  kinds  moves  through  a  solution  under  the  in- 
fluence of  a  current  of  a  given  electromotive  force.  Since 
i  equivalent  of  ionized  matter  will  transport  96,540 
coulombs  of  electricity,  and  A  is  the  amount  of  current 
carried  in  i  second  between  electrodes  i  cm.  apart  when 
the  electromotive  force  is  i  volt  (for  current  strength 

voltage  i        \ 

=  — r— ,   and    conductivity  =  — r- I ,    the    term 

resistance  J     resistance/ 

must  represent  the  fraction  of  i  centimeter  trav- 

96540 

ersed  by  the  two  kinds  of  ionized  matter  in  i  second,  i.e., 
the  sum  of  the  distances  traversed  by  each.  We  have, 
then, 


96540 

and,  knowing  the  relative  mobilities,  we  can  find  the 
absolute  mobilities  of  the  two.  An  illustration  of  this  is 
given  by  a  o.oooi  molar  solution  of  KC1,  which  at  18° 
gives  -4  =  128.9.  We  obtain,  consequently, 

128.9 

:  0.001345  cm.  per  second, 


and  since  from  Hittorfs  results  .£'  =  49  when  Cl'  =  5i 
(p.  179,),  the  absolute  mobility  of  K'  is  0.00066  cm.  per 
second  and  of  Cl'  is  0.00069,  when  the  potential  gradient 
is  i  volt  and  the  solution  is  o.oooi  molar. 

Knowing  the  absolute  mobility  of  the  various  kinds  of 
ionized  matter,  then,  we  can  calculate  the  equivalent 
conductivity  at  infinite  dilution,  or  at  any  other  dilution 
provided  we  know  a,  for  we  have  AM  =  (va +7^)96540, 
and  Av = a (va +vc)g6 540,  where  the  va  and  vc  refer  to 


186  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

the  mobilities  in  centimeters  per  second.  In  the  table 
below  are  some  of  the  absolute  mobilities  as  given  by 
Kohlrausch. 

ABSOLUTE  VELOCITY  OF  IONIZED  MATTER  AT  18°  IN  CMS.  PER  SECOND. 


K'        = o . 00066 

NH;    =0.00066 

Na*  =  o .  00045 
Li'  =0.00036 
Ag*  =0.00057 
Cr2O7"=  0.000473 


H'  =0.00320 
Cl'  =  o .  00069 
NO3'  =  o .  00064 
C103'=  0.00057 
OH'  =  0.00181 
Cu"  =0.00031 


Not  only  can  these  values  be  calculated  in  this  way, 
but  they  can  also  be  directly  observed  by  experiment. 
The  method,  as  used  by  Whetham,*  depends  upon  the 
speed  with  which  a  color-boundary  between  two  equally 
dense  solutions  moves  during  the  application  of  the  cur- 
rent. Imagine  two  solutions  which  contain  a  common  and 
colorless  kind  of  ionized  matter,  but  which  are  colored 
differently,  and  designate  the  salts  as  AC  and  EC.  The 
passage  of  the  current  will  thus  cause  ionized  C  to  go  in 
one  direction  and  ionized  B  and  ionized  A  to  go  in  the 
other.  Provided  these  solutions  are  placed  in  a  horizontal 
tube  so  that  before  the  current  is  applied  the  color-boun- 
dary will  be  quite  sharp,  the  application  of  the  current 
will  cause  the  boundary  to  advance  with  a  speed  which 
depends  upon  the  potential  gradient,  and  which  can  be 
measured.  This  speed,  however,  will  be  that  of  the 
colored  ionized  matter,  for  that  produces  the  color.  In 
the  table  on  the  opposite  page  are  given  the  mobilities  as 
found  by  calculation  and  by  direct  experiment,  and  but  a 
glance  is  necessary  to  convince  one  of  the  correctness 
of  our  conclusions  as  to  electrical  conduction  in  solution : 

*  Phil.  Trans.,  i893A,  337;    1895^  507. 


ELECTROCHEMISTRY.  1.87 

COMPARISON  OF  ABSOLUTE  IONIC  VELOCITIES. 

Kind  of  Whetham,  Kohlrausch, 

Ionized  Matter.  by  Experiment.  Calculated. 

Cu"  0.00026  0.00031 

0.000309 

Cl' 0.00057  0.00069 

o . 00059 
0.00047 

Cr2O7" 0.00048          *  0.000473 

o . 00046 

Empirical  relations. — There  are  two  empirical  rela- 
tions with  regard  to  electrical  conductivity  which  are 
often  very  useful.  The  first  of  these  is  of  particular  value 
to  the  chemist  as  a  means  of  determining  the  constitu- 
tion of  chemical  substances,  while  the  second  applies 
directly  to  the  electrical  behavior  of  strong  (i.e.  largely 
ionized)  electrolytes. 

Although  the  Ostwald  dilution  law, 


K  = 


holds  for  all  monobasic  organic  acid,  and  for  dilutions 
below  that  at  which  0^  =  50%  for  all  polybasic  ones. 
Ostwald  found  that  the  difference  between  the  equivalent 
conductivity,  oj  the  sodium  salt  of  any  organic  acid,  at 
the  dilution  7=1024  and  that  at  F  =  32,  is  approximately 
equal  to  nXio  units,  where  n  is  the  basicity  of  the  acid. 
Thus  formic  acid  shows  a  difference  of  10.3  units,  qui- 
ninic  of  19.8,  pyridintetracarboxylic  of  41.8,  and  pyridin- 
pentacarboxylic  of  50.1. 

The  other  relation  has  already  been  mentioned  (p.  145). 
It  enables  us  to  find  the  equivalent  conductivity  of  a 
neutral  salt  at  one  dilution,  provided  we  know  that  at 
another,  and  the  salt  is  considerably  ionized,  i.e.,  when 


188  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

Av  is   not    very   different    from  A^.    The    relation  ob- 
served is  as  follows: 


or  <x>  =  ni-n2'Cv+vt 

where  n\  and  n2  are  the  valences  of  the  anion  and 
cathion  matter  respectively,  and  cv  is  a  constant  for  all 
electrolytes.  When  cv  is  known  for  all  dilutions,  and 
also  the  terms  Av,  n\,  and  n2,  we  can  find  the  value  of 
A^,  i.e.,  the  equivalent  conductivity  at  infinite  dilution. 
If  we  designate  (HI  -n2  -cv)  by  dv,  then 


Below  are  given  the  values  of  dv  for  different  dilutions 
and  values  of  n\  -n2  at  25°. 

Valence,  »i  "nz  <f64  di2&  dz5&  ^512  ^1024 

I II  8  6  4  3 

2 21        l6        12         8         6 

3 30  23  17  12  8 

4 42  31  23  l6  i° 

5 53  39  29  21  13 

6 60  48  36  25  16 

This  behavior  may  be  summed  up  in  words  as  follows  : 
The  decrease  of  equivalent  conductivity  is  roughly  con- 
stant for  salts  of  the  same  type,  and  the  decrease  in  equiva- 
lent conductivity  for  salts  of  different  types  is  proportional 
to  the  product  of  the  valences  of  the  kinds  of  ionized  matter 
present. 

The  ionization  of  water. — Water  ionizes  to  a  very  slight 
extent  into  H*  and  OH'.  The  specific  conductivity  of 
an  especially  pure  sample,  as  determined  by  Kohlrausch, 
is  0.014X10-6  at  o°,  0.040X10-6  at  18°,  0.055  Xio"6 
at  25°,  0.084X10-6  at  34°,  and  0.17X10-6  at  50°, 


OF 


ELECTROCHEMISTRY.  189 

N^L'FftPN^ 

From  this  conductivity,  naturally,  we  can  calculate 
the  degree  of.  ionization,  provided  we  know  the  ionic 
conductivities  of  H"  and  OH'  at  that  temperature.  Since 
i  mole  of  ionized  H*  has  a  conductivity  of  318  at  18°, 
and  i  mole  of  OH'  174  uhits,  water,  if  completely  ion- 
ized, would  give  a  molar  conductivity  of  492  units.  As 
the  specific  conductivity  at  18°  is  0.04  Xio~6,  that  of  a 
liter  would  be  0.04  X 10  ~6  X  io3 ;  hence 

0.04  X 10  ~3 

-  = 
492 

which  is  the  concentration  of  ionized  H'  and  of  ionized 
OH'  in  i  liter  of  water,  or,  in  other  words,  there  are 
17  grams  of  ionized  OH'  and  i  gram  of  ionized  H*  in 
12,000,000  liters  of  water. 

It  is  to  be  remembered  here  that  492  is  the  value  which 
would  be  given  if  i  mole  of  ionized  H*  and  i  mole  of 
ionized  OH'  were  present  together,  between  electrodes 
i  cm.  apart  and  large  enough  to  contain  between  them 
the  1 8  grams  of  water,  or  any  other  volume  containing 
i  mole  each  of  H"  and  OH'.  The  calculation  of  the 
ionization  is  usually  made  for  i  liter,  since  that  is  the 
volume  to  which  we  make  up  solutions. 

The  solubility  of  difficultly  soluble  salts. — When  a 
saturated  solution  of  any  so-called  insoluble  salt  is  so 
dilute  that  we  may  assume  complete  ionization,  ( A 
we  have 


and 

i      /cXio3 
i.e.,  c  =  T?  =  —A —  moles  per  liter. 


IQO  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

This  method  is  exceedingly  satisfactory,  so  long  as  the 
specific  conductivity  of  the  saturated  solution  differs 
sufficiently  from  that  of  water,  and  we  can  safely  assume 
the  degree  of  ionization  to  be  i  or  determine  it.  The 
calculation  of  A»  must  naturally  be  made  by  aid  of 
the  results  on  page  184,  for  by  experiment  it  would  be 
impossible  for  us  to  find  the  term  A*>  without  knowing  the 
solubility  of  the  salt.  In  case  the  conductivities  of  the 
constituents  of  the  salt  for  which  we  are  to  determine 
A»  are  not  included  in  the  table  we  can  naturally  calcu- 
late them  from  the  values  at  infinite  dilutions  of  salts  for 
which  Joo  can  be  found  by  experiment.  In  general,  then, 
we  would  have 


Thus  for  BaSO4,  we  have 


i.e.,  Joo(|BaSO4)  =  1  15  +  128  -  115  =  121. 

The  influence  of  temperature  upon  conductivity  is  pri- 
marily the  result  of  temperature  upon  the  speed  of  migra- 
tion (for  the  ionization  does  not  change  very  greatly  with 
the  temperature,  see  page  144)  and  is  shown  in  the  table 
on  page  184.  In  general  the  variation  in  the  equivalent 
conductivity  of  largely  ionized  salts  is  about  2^%  per  de- 
gree, and  this,  naturally,  must  always  be  considered  when 
calculating  ionization  or  solubility,  as  was  done  above. 

For  the  calculation  of  the  conductivity  of  a  mixture 
of  substances  the  reader  must  be  referred  elsewhere,*  for 
the  relations  are  too  complex  to  be  discussed  here. 

*  "Elements,"  pp.  396-399. 


ELECTROCHEMISTRY.  19* 


ELECTROMOTIVE   FORCE. 

The  chemical  or  thermodynamical  theory  of  the  cell. — 

We  shall  not  consider  here  either  the  methods  for  deter- 
mining the  electromotive  force  nor  the  standard  cells 
upon  which  such  measurements  are  based,  but  shall 
devote  ourselves  exclusively  to  the  consideration  of  those 
factors  which  condition  the  rise  and  magnitude  of  an 
electromotive  force  in  a  system. 

Since  in  general  it  is  the  chemical  energy  of  a  process 
which  is  transformed  into  electrical  energy,  and  since 
the  heat  developed  by  a  chemical  reaction  under  certain 
conditions  is  proportional  to  the  chemical  energy  involved, 
it  is  possible  to  derive  a  formula  from  which  the  electrical 
energy  can  be  calculated  when  the  heat  developed  by 
the  reaction  is  known.  Such  a  formula,  however,  proves 
to  be  satisfactory  only  in  isolated  cases,  and  the  varia- 
tion has  been  shown  to  be  due  to  the  loss  or  gain  of 
heat  during  the  process,  i.e.,  either  less  or  more  than  the 
heat  developed  by  the  chemical  reaction  itself  is  trans- 
formed into  electrical  energy.* 

Imagine  a  reversible  cell  in  which  the  amount  of  heat 
q  is  liberated  or  absorbed  during  the  passage  of  i  gram 
equivalent  of  ionized  matter  through  the  solution.  As- 
sume this  cell  to  be  in  a  constant  temperature  bath  so 

*  To  obtain  electrical  energy  from  a  chemical  reaction  it  is  usually 
necessary  to  so  separate  the  process  that  it  may  take  place  in  two  por- 
tions, at  points  which  are  spatially  separated.  Thus  zinc  dissolves  in 
acid,  giving  off  hydrogen  gas  and  evolving  heat.  When  the  zinc  is 
connected  by  a  wire  to  a  plate  of  platinum,  however,  and  both  are 
placed  in  acid,  the  zinc  dissolves,  but  the  hydrogen  is  evolved  from 
the  platinum  plate  and  a  current  of  electricity  flows  through  the  wire; 
the  process  has  been  separated  into  two  spatially  separated  portions, 
and  a  current  is  the  result. 


192  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

that  its  temperature  cannot  vary  —  i.e.  if  heat  is  absorbed 
it  is  replaced,  if  liberated  it  is  removed  —  thus  preventing 
q  from  causing  any  change  in  the  temperature  of  the  cell. 
If  TT,  the  electromotive  force  of  this  cell,  is  just  compen- 
sated by  —  TT,  the  process  will  be  in  equilibrium.  For 
two  energies  in  equilibrium,  however,  we  have  found 
(p.  15)  that 


where  c  and  i  are  the  factors  of  the  one  kind  of  energy, 
and  ci  and  i\  those  of  the  other.  In  this  case  the  two 
kinds  of  energy  are  electrical  and  thermal,  hence 


or 


which  gives  the  change  in  E.M.F.  due  to  the  absorption 
or  liberation  of  q  calories  during  the  process. 

If  we  now  pass  96,540  coulombs  of  electricity  through 
such  a  cell,  during  which  to  keep  the  temperature  con- 
stant it  is  necessary  to  supply  q  calories,  the  electrical 
energy  obtained  by  the  process  must  be  equal  to  the 
chemical  energy  involved  plus  the  electrical  energy  equiv- 
alent to  the  heat  q.  We  have,  then,  expressing  all  terms 
in  like  units, 


*  According  to  the  second  law  of  thermodynamics  (p.  48),  the  capacity 
factor  of  heat  energy  is  equal  to  ~,  for  we  find  AQ=-j=JT,  and  know 
that  T  is  the  intensity  factor. 


ELECTROCHEMISTRY.  193 

or,  since  Eg  =  ne0, 


TT= 


i.e.,  the  actual  E.M.F.,  nt  is  only  equal  to  that  calculated 
from  the  chemical  energy  (as  heat)  of  the  process  when 
the  E.M.F.  is  independent  of  the  temperature.  Other- 
wise TT  is  smaller  or  larger  than  — ,  according  as  T-r^ 

£Q  A 1 

is  negative  or  positive  in  value. 

An  illustration  of  the  application  of  this  formula 
is  furnished  by  the  Grove  gas-cell.  Here  ;r=  1.062 
volts  and  g,  the  heat  evolved  by  the  chemical  reaction, 

34200        An 
is    34,200     calories,    hence    1.062= ^    ~AT>     Le'» 

TAir 

-T=r  =  —0.418  volts,  in  place  of   —0.416  as  found  by 

actual  experiment.  The  value  23,110  here  is  the  quan- 
tity 96,540  coulombs  expressed  in  calories,  i.e.,  is 
96, 540X0. 2394  XTT  =  23,1  IOTT.  In  other  words,  23, HOT:  is 
the  work  in  calories  necessary  to  separate  i  gram  equiv- 
alent of  any  substance  at  n  volts. 

The  osmotic  theory  of  the  cell. — Considering  a  cell 
from  the  standpoint  of  our  conclusions  respecting  the 
nature  of  electrolytes,  it  is  possible  to  see  more  clearly 
into  the  cause  of  the  rise  of  a  difference  in  potential 
between  two  solutions,  or  a  metal  and  a  solution. 

Assume  we  have  two  solutions  in  contact,  and  that  they 
contain  the  same  kind  of  monovalent  ionized  matter  in  dif- 
fering concentration.  The  difference  of  potential  existing 
on  their  boundary  can  now  be  calculated  by  aid  of  the  fol- 
lowing process  of  reasoning :  If  Ua  and  Uc  are  the  mobilities 
of  the  respective  kinds  of  ionized  matter,  then,  by  the  pas- 


194  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

sage  of  96,540  coulombs  of  electricity,  the  following  changes 
must  take  place  :  Assuming  that  the  current  enters  on  the 
concentrated  side  and  passes  through  both  solutions, 

C      gram  equivalents  (moles  in  this  case)  of  posi- 


tively  charged  ionized  matter  will  go  from  the  concen- 
trated side  to  the  dilute,  and,  during  the  same  time, 

T,    "TJ  gram  equivalents  of  negatively  charged  ionized 

L/C  i    Ua 

matter  will  go  from  the  dilute  solution  to  the  other.  Let 
p  be  the  osmotic  pressure  of  the  two  kinds  of  ionized 
matter  in  the  concentrated  solution  and  pf  that  of  those 
in  the  dilute  solution,  the  maximum  osmotic  work  to  be 
done  by  the  process,  then,  will  be  (p.  99) 


and  this  must  be  equal  to  TTSQ  for  the  process  going  in 
this  way,  i.e.,  to  the  electrical  work  done  at  the  contact 
surface  of  the  two  solutions.  Since  R  in  calories,  when 
divided  by  23,110,  gives  the  value  in  electrical  units,  we 
can  find  in  this  way  the  difference  of  potential  of  two 
solutions,  and  experiment  has  shown  the  calculated  values 
to  agree  with  those  observed. 

This  same  method  of  reasoning  may  be  applied  to  the 
cell 


Concentrated  amalgam 
of  the  metal  M . 


Water  solution  of  a  salt 
of  the  metal  M . 


Dilute  amalgam  of  the 
metal  M. 


In  this  case  the  passage  of  99)540  coulombs  causes 
i  gram  equivalent  of  the  metal  M  to  go  from  the  con- 
centrated side  to  the  dilute,  and,  if  there  are  n  equivalents 
to  the  mole  (i.e.,  the  metal  is  n  valent),  the  maximum 


ELECTROCHEMISTRY.  195 

work  per  equivalent  will  be 


where  ci  and  c2  are  the  concentrations  of  the  metal  M 
in  the  two  amalgams.  Again,  here,  the  value  of  TT  in 
volts  can  be  found  by  dividing  the  expression  by  23,110. 
And  in  the  same  way  we  can  calculate  a  formula  for 
a  cell  with  electrodes  of  the  same  soluble  metal  in  two 
different  concentrations  of  a  solution  of  a  salt  or  salts 
of  that  metal.  Consider,  for  example,  the  cell 


Cu 


dilute 
CuSO, 


concentrated 
CuSO4 


Cu. 


By  passing  a  current  through  this  cell  in  the  direction 
of  the  arrow  the  following  changes  will  take  place: 

1.  For   each   96,540   coulombs   of   electricity    i    gram 
equivalent  of  copper  will  dissolve  from  the  electrode  in 
the  dilute   solution,  i.e.,  will  be   transformed   from  the 
metallic  to  the  ionic  state; 

2.  At  the  boundary  of  the  two  solutions  the  process 
described  above  will  take  place;  and 

3.  One  gram  equivalent  of  ionized  Cu"  will  be  de- 
posited from  the  concentrated  solution  upon  the  electrode. 

As  the  result  of  processes  (i)  and  (3)  i  gram  equiva- 
lent of  ionized  Cu"  will  go  from  the  concentrated  to  the 
dilute  solution.  The  maximum  work  of  this  process, 
then,  for  each  96,540  coulombs,  will  be 

RT 


where  n  is  the  valence  of  the  metal  and  pi  and  p2  are  the 


196  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

osmotic  pressures  exerted  by  the  ionized  Cu"  in  the  two 
solutions. 

By  the  second  process  (contrast  with  direction  of  cur- 
rent in  the  case  above)  we  have  as  the  work 


P 


n 


Neglecting   the    difference    of   potential   between   the 
liquids,  which  is  usually  very  small,  we  have 


or,  including  that, 

RT     2U 


_ 


Uc+Ua 


i.e.  the  sum  of  the  two,  where,  when  the  value  2  calories 
is  substituted  for  R,  and  23,100  for  £0,  TT  is  given  in  volts. 


< 

By  separating   the   amount   of  work  -—log        into 

ns0  p2 

two  portions,  so  that  each  will  represent  the  maximum 
work  at  an  electrode,  we  can  write,  neglecting  the  dif- 
ference of  potential  at  the  boundary  of  the  two  solutions, 

RT         P     RT         P 


where  P  is  a  constant  for  any  one  metal  at  one  tempera- 
ture in  the  same  solvent,  and  is  called  the  electrolytic 
solution  pressure. 

Here,   again,  it  is  not  so  much  a  question  of  what 
electrolytic  solution  pressure  is,  as  it  is  of  what  we  mean 


ELECTROCHEMISTRY.  197 

by  the  word  electrolytic  solution  pressure.  It  will  be 
seen  that  this  conception  leads  to  a  constant  value  for 
any  one  metal,  and  a  value  for  other  metals  which  can 
be  found  in  terms  of  the  first  by  finding  the  electromotive 
force  when  the  solutions  contain  the  same  quantity  of 


T) 

ionized  metal,  for    this   is    then  equal  to  -    -  log  — 

neo        '  J^2 

We  speak  of  positive  electrolytic  solution  pressure  when 
the  metal  dissolves,  negative  when  the  ionized  metal 
deposits  upon  it. 

If  a  metal  has  a  tendency  to  dissolve  in  a  solution, 
i.e.  to  form  ionized  metal,  the  solution  must  be  positive 
against  it,  for  the  metal  loses  positive  electricity.  Un- 
less there  is  some  means  of  neutralizing  this  difference 
of  potential,  it  is  quite  evident  that  solution  must  soon 
cease,  and  that  all  the  positively  charged  ionized  matter 
present  (and  remaining  as  such)  must  be  immediately  at- 
tracted back  again  to  the  metal.  Naturally,  if  the  negative 
charge  on  the  plate  becomes  neutralized  by  a  positive 
charge  from  without  the  system,  solution  will  continue 
until  the  electrolytic  solution  pressure  is  compensated  by 
the  osmotic  pressure  of  the  ionized  metal  in  the  solution, 
or  until  the  metal  is  all  dissolved.  This  is  the  case,  for 
example,  with  zinc,  where  the  electrolytic  solution  pres- 
sure is  positive  and  has  a  very  high  value. 

The  other  extreme,  i.e.  where  ionized  metal  from  the 
solution  is  usually  precipitated,  i.e.  is  transformed  to 
the  un-ionized  metallic  state,  is  illustrated  by  copper  in 
its  solutions.  Here  ionized  metal  is  precipitated  upon 
the  electrode,  which  thus  acquires  a  positive  charge. 
Naturally,  here  also,  the  amount  deposited  is  exceed- 
ingly small,  for  it  also  produces  a  difference  of  potential; 
and  when  the  positive  charge  is  neutralized  by  a  negative 


198  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

one  from  without  the  system,  the  process  of  precipita- 
tion continues. 

By  combining  the  electrodes  of  two  such  systems,  i.e. 
the  zinc  and  copper  by  a  wire,  the  solutions  by  a  siphon, 
the  process  in  each  may  continue,  for  the  charges  upon 
the  electrodes  can  neutralize  and  consequently  give  rise 
to  a  current  which  flows  until  the  zinc  is  all  dissolved 
or  the  ionized  copper  all  precipitated,  with  a  loss  of  its 
charge,  as  metal.  Further,  in  such  a  case,  by  applying 
a  positive  current  to  the  copper  electrode,  it  is  possible, 
if  the  impressed  electromotive  force  be  greater  than  that 
of  the  cell,  to  reverse  the  process,  i.e.,  to  dissolve  copper 
and  precipitate  zinc.  The  action  of  such  a  cell,  when 
in  operation,  then;  is  to  transform  metallic  zinc  into  the 
ionized  metal,  and  ionized  copper  into  un-ionized  metal. 

The  condition  of  the  zinc  and  copper  electrodes  before 
they  are  connected  is  shown  in  the  figure  below.  To 
obtain  an  E.M.F.  from  them  it  is  only  necessary  to  con- 
nect the  solutions  by  a  siphon  and  the  electrodes  through 
a  wire. 


From  the  formula  on  page  196  it  will  be  seen  that  it  is 

p 

the  ratio  —  which  is  of  importance.     The  osmotic  pres- 
sure pj  then,  has  much  to  do  with  the  size  of  this  ratio. 


ELECTROCHEMISTR  Y. 


199 


This  fact  is  observed  best  by  the  addition  of  potassium 
cyanide  to  the  copper  solution  in  which  there  is  a  copper 
electrode.  Here  experiment  (loss  of  color,  for  example) 
shows  the  formation  of  the  ionized  complex  CuCN/' 
from  the  ionized  Cu",  and  at  the  same  time  it  is  observed 
that  the  copper  electrode  dissolves,  i.e.  becomes  negative 
in  value  (all  ionized  Or*  is  removed  as  it  is  formed), 
so  that  P  appears  positive  in  value.  Indeed,  the  addition 
of  potassium  cyanide  to  the  copper  side  of  such  a  com- 
bination of  a  copper  and  zinc  system,  for  this  reason, 
reverses  the  polarity  of  the  cell,  and  the  copper  becomes 
the  negative  pole. 

The  actual  presence  of  such  a  layer  of  ionized  matter 
around  the  oppositely  charged  metal  (the  Helmholtz 
double  layer,  as  it  is  called),  as  we  have  concluded  must 
necessarily  be  present,  has  been  shown  by  Palmaer* 
with  an  arrangement  which  in  principle  is  like  that  shown 
in  the  figure  below. 


Drops  of  mercury  are  allowed   to  fall  into  a  weak 
solution  containing  ionized  mercurous  mercury,  metallic 

*  Wied.  Ann.,  28,  257,  1899. 


200  PHYSICAL  CHEMISTRY  FOR.  ELECTRICAL  ENGINEERS. 

mercury  being  in  the  bottom  of  the  tube  containing  the 
solution.  If  now  the  double-layer  theory  is  true,  the 
drops  of  Hg  as  they  form  should  have  the  electricity  of 
the  ionized  Hg2  deposited  upon  them,  and  these  positively 
charged  drops  should  then  attract  the  negatively  charged 
ionized  matter,  forming  on  each  a  double  layer.  When 
such  a  drop  reaches  the  mercury  at  the  bottom  it  will 
unite  with  that,  forming  ionized  Hg2  once  more  and 
releasing  the  ionized  radical,  and  the  concentration  of 
mercury  salt  should  be  greater  at  the  bottom  than  at  the 
top;  and  Palmaer's  experiments  showed  this  difference 
in  concentration  to  actually  exist. 

Experiment  has  shown  that  the  metals  Na,  K .  .  .  , 
etc.,  up  to  Zn,  Cd,  Co,  Ni,  and  Fe  are  always  negative 
against  their  solutions,  i.e.,  P>p. 

The  noble  metals,  on  the  contrary,  are  positive  against 
their  solutions,  although  in  some  few  cases  it  is  possible 
to  get  a  solution  in  which  P>p.  In  general,  though, 
for  the  noble  metals  P  <  p. 

A  negative  element  has  exactly  the  same  action  except 
that,  in  general,  as  far  as  is  known,  P>p.  Here,  although 
P>p,  the  electrode  is  positive  against  the  solution,  for 
the  negatively  charged  ionized  matter  formed  from  the 
electrode  leaves  positive  electricity  behind. 

In  general,  the  electrolytic  solution  pressure  depends 
upon  the  temperature,  the  nature  of  the  solvent,  and 
the  concentration  of  the  substance  in  the  electrode  (see 
p.  206). 

Naturally,  from  this  experimental  conception  of  electro- 
lytic solution  pressure  it  is  possible  to  derive  the  formula 
for  the  E.M.F.  given  by  any  combination.  When  i 
mole  of  ionized  metal  is  formed  from  an  electrode  against 
the  osmotic  pressure  p,  the  osmotic  work  is 


ELECTROCHEMISTRY.  201 


from  which,  by  integration,  we  obtain 


The  corresponding  electrical  work,  however,  is  7re0, 
where  TT  is  the  difference  of  potential  and  SQ  is  the  quantity 
of  electricity  carried  by  i  gram  equivalent  of  ionized 
matter.  We  have  then 

g—, 

RT1       P 

«-—  log,-, 

from  which  when  P  =  P,  n  —  o.  Expressing  this  in  elec- 
trical units  we  find,  in  general,  where  n  is  the  valence 
of  the  metal, 


0.0002  P 

7T=-          -T  log -VOltS, 


so  that  at  if 


For  a  substance  forming  negatively  charged  ionized 
matter  we  have,  correspondingly, 

0.0002  _         P    0.0002  _ 


202   PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

Combining  two  electrodes,  knowing  that  ionized  matter 
is  formed  at  one  and  disappears  at  the  other,  we  have 

0.0002 .      PI     0.0002  P2     u 

*- <*>  -**> = —  logh '  ~^r  T  g  h volts> 

Differences  of  potential.  Calculation  of  the  electro- 
lytic solution  pressure. — In  order  to  measure  the  electro- 
lytic solution  pressure,  i.e.  P  in  the  equation 


0.0002 


when  p,  the  osmotic  pressure  of  the  ionized  metal,  is 
known,  it  is  necessary  first  to  determine  TT,  the  difference 
of  potential  between  the  metal  and  its  solution.  To 
do  this,  naturally,  it  is  necessary  to  combine  the  electrode 
with  another,  which  gives  a  known  difference  of  poten- 
tial, and  thus,  knowing  n  and  n\  (see  above),  we  can  readily 
find  7T2.  It  has  been  found  that  when  mercury  drops  into 
an  electrolyte  the  difference  of  potential  soon  becomes 
zero.  Any  cell  of  which  this  arrangement  is  one  elec- 
trode, then,  gives  as  its  electromotive  force  the  difference 
of  potential  existing  between  the  metal  and  solution  at 
the  other  electrode.  As  this  dropping  electrode  is  cum- 
bersome and  inconvenient  for  general  use,  it  is  usual  to 
employ  the  so-called  normal  (or  tenth-normal)  electrode 
for  the  purpose,  its  value  being  determined  once  for  all 
against  the  dropping  electrode.  The  electrode  in  com- 
mon use  is  made  up  of  metallic  mercury  in  a  molar 
(or  o.i  molar)  solution  of  potassium  chloride  which  is 
saturated  with  calomel.  The  value  of  the  normal  elec- 


ELECTROCHEMISTRY.  203 

trode  *  is  —0.56  volt  at  18°,  i.e.,  the  solution  (to  which 
the  sign  always  refers)  is  negative  by  0.56  volt  against 
the  metal.  The  value  of  the  electrode  when  o.i  molar 
salt  is  used  is  —0.613  at  18°,  for  mercury  has  a  negative 
electrolytic  solution  pressure,  and  the  weak  solution  of 
KC1  dissolves  a  greater  amount  of  calomel,  i.e.,  con- 
tains a  greater  amount  of  ionized  mercury. 

Since  p,   the  osmotic  pressure  of  the  ionized   metal 
in  atmospheres,  is  equal  to  the  product  of  the  concen- 

T 

tration  in  moles  per  liter  and  22.4  —  ,  we  can  readily 

273 
calculate  P  in  the  equation 


RT 


if  we  know  TT,  the  difference  of  potential  between  the 
metal  and  its  solution  at  the  absolute  temperature  T. 
At  17°  C.,  thus,  we  have 


from  which  the  value  of  P  in  atmospheres  can  be  found. 
The  values  of  P  for  the  various  metals,  obtained  in 
this  way,  are  given  in  the  table  on  the  following  page. 
It  is  to  be  remembered  here  that  these  values  are  merely 
symbolical,  for  the  gas  laws  may  not  be  applied  to  such 
an  extent.  The  relation  between  these  numbers,  how- 
ever, are  those  that  would  be  found  if  the  E.M.F.'  were 
measured  under  the  .condition  that  the  osmotic  pressures 
of  the  ionized  metals  were  the  same. 

*  For  details  as  to  this,  see  "  Elements,"  pp.  417-421. 


204  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

ELECTROLYTIC  SOLUTION  PRESSURES  OF  THE  METALS. 
Zinc  ..........................   9-QX  io18  atmospheres 

Cadmium  .....................   2  .  yX  io8 

Thallium  ......................   7  .  ;X  io2 

Iron  ..........................    i.aX  io4 

Cobalt  ........................    i  .pX  10° 

Nickel  ........................    i  .  3X  10° 

Lead  .........................    i.iXio-3 

Hydrogen  .....................   9-9X  io~4 

Copper  ........................   4.8X  io-20 

Mercury  ......................    i.iX  io~16 

Silver  .........................   2.3X  io~17 

Palladium  ____  .,  ................    i  .  5  X  io~38 

When  the  metals  acting  as  electrodes  are  inert,  as  in 
the  so-called  oxidation  and  reduction  cells,  the  significance 
of  the  conception  of  electrolytic  solution  pressure  natu- 
rally disappears.  An  example  of  such  a  cell  is  given  by 
the  arrangement 

plat.  Pt—  FeCl3  sol  ____  SnCl2  sol.—  plat.  Pt. 

Here  it  is  simply  a  question  of  the  electric  charge  upon 
the  ionized  matter,  and  the  action  may  be  expressed 
by  the  equation 


The  heat  of  ionization.  —  Knowing  the  difference  of 
potential  existing  between  a  metal  and  a  solution,  and 
its  rate  of  change  with  a  variation  in  the  temperature, 
it  is  possible  from  the  formula  on  page  192  to  find  the 
heat  of  ionization  of  the  metal.  We  found  there  that 

fp 

An 
or  ?r=  --  \-T  —  . 


ELECTROCHEMISTRY.  205 

The  term  Ec,  here,  is  the  heat  produced  when  i  gram 
equivalent  of  the  metal  goes  from  the  metallic  to  the 
ionic  state,  and  its  value  is  given  by  the  formula 


E 


But  (page  174), 

eon  =  96540  X 0.2394  XTT  volts  =  231  ion  cals. ; 

hence  the  heat  of  ionization  EC1  for  i  gram  equivalent, 
can  be  found  from  the  relation 

JTT\ 

cals. 


For  copper  in  copper  acetate  (molar)  at  17°,  ;r=o.6, 

i   ^  i  -i     An  . 

and  ~TJ^  —  0.0007  74>  while  -j~  for  copper  in  copper  sul- 

.    An 
phate    is    0.000757,    i.e.    an    average  value    of   -j=,  of 

0.000766  volt,  hence  Ec,  the  heat  of  ionization  of  copper, 
is  8736  cals.  per  gram  equivalent,  or  17,472  cals.  per 
mole.  It  was  in  this  way  that  the  value  for  H  was  de- 
termined for  use  in  the  table  given  on  page  95. 

Concentration  cells. — If  the  electrodes  of  a  cell  are 
amalgams  containing  different  concentrations  of  the  same 
metal,  and  the  solutions  are  identical  with  respect  to  the 
ionized  metal,  our  general  formula  for  the  electromotive 
force  (p.  202), 

0.0002^        PI     0.0002^        P-2, 

7T  =  -        -rlog-^--          -T  log  -=•  VOltS, 
tti  3   pi  112  °  P2 


206  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 
becomes,  since  n\  =  n^  and  pi  =  p2, 
0.0002  PI 


where  PI  and  P2  are  the  electrolytic  solution  pressures 
of  the  two  amalgams  with,  respect  to  the  dissolved  metal. 
We  have,  then,  a  concentration  cell  in  which  the  electrodes 
have  different  concentrations.  Since  the  electrolytic  solu- 
tion pressure  due  to  the  dissolved  metal  is  proportional 
to  the  amount  of  this  dissolved  in  the  mercury,  the  for- 

mula acquires  the  simpler  form  TT=—      -  Tlog  —  volts. 

This  equation  has  been  tested  experimentally  for  zinc 
and  copper  amalgams  and  found  to  express  "very  accu- 
rately the  relations  observed. 

Zinc  in  amalgams,  as  well  as  most  other  metals,  exists 
in  monatomic  form,  i.e.,  the  formula  weight  (by  freezing- 
point  definition,  for  example)  is  found  to  be  identical 
with  the  combining  weight.  If  it  were  diatomic,  i.e. 
the  formula  weight  contained  two  combining  weights, 
the  above  formula  would  have  assumed  a  different  form. 
For  the  movement  of  the  same  weight  of  ionized  matter 
(see  page  201)  the  osmotic  work  would  then  have  been 

JJ^Tlog  —  ,  and,  since  the  electrical  work  would  have 

C2 

been  unchanged,  i.e.  2X9654071,  we  should  have  had 

1  0.00002  Ci 

it  =  —  ~T  log  —  volts, 

2  n  b  c2 

i.e.,  the  electromotive  force  would  have  been  one-half 
what  it  has  been  found  to  be, 


ELECTROCHEMISTRY.  207 

Another  example  of  a  concentration  cell  due  to  a 
different  concentration  in  the  electrodes  is  given  by  cells 
of  the  type  of  the  Grove  gas-battery  in  an  altered  form. 
The  .  electrodes,  here,  are  of  platinized  platinum,  in 
which  the  gas  is  absorbed  under  different  pressures,  and 
are  placed  partly  in  a  liquid  and  partly  in  the  gas  at 
a  corresponding  partial  pressure.  Such  an  electrode  is 
to  be  considered  as  a  perfectly  reversible  gas  electrode,* 
i.e.,  one-  from  which  the  material  absorbed  as  a  gas  is 
given  up  in  the  ionized  state,  for  the  metal  acts  simply 
as  a  conductor,  as  has  been  shown  experimentally  by 
the  use  of  different  metals,,  the  same  result  being  always 
obtained.  In  this  way  reversible  gaseous  electrodes  of 
all  kinds  can  be  made.  Oxygen  as  an  electrode,  how- 
ever, gives  off  ionized  OH',  since  ionized  O"  is  not 
known  to  exist,  and  forms  O  and  H2O  when  the  ionized 
OH'  gives  up  its  charge  to  it. 

If  we  have  two  electrodes  of  H,  under  different  pres- 
sures, in  contact  with  a  liquid  containing  ionized  H',  we 
shall  obtain  a  certain  E.M.F.  This  may  be  calculated 
in  two  ways,  as  we  did  in  the  case  of  amalgams.  In 
the  second  way,  however,  the  process  is  slightly  different, 
since  one  mole  of  H  gas  forms  two  moles  of  ionized  H* 

T> 

(p.  84).     The  osmotic  work  is  equal  to  RT  log^  -^,   as 

*  2 

before.     The    electrical    work,    however,    which    corre- 
sponds to  this  is  2eox,  for  H2  =  2H';    hence 

RT        Pi 


*  The  electrolytic  solution  pressure  here  of  the  gas  electrode  is  pro- 
portional to  the  wth  root  of  the  gaseous  pressure,  where  n  is  the  number 
of  combining  weights  in  one  formula  weight  of  gas. 


208  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

i.e.,  we  have  2  in  the  denominator,  notwithstanding  the 
fact  that  the  gas  is  monovalent. 

When  the  electrodes  of  the  cell  are  of  the  same  metal, 
but  the  concentration  of  ionized  metal  in  solution  differs 
on  the  two  sides,  we  have  the  typical  form  of  a  concen- 
tration cell.  An  example  of  this  arrangement  is 

Ag(AgN03  cone.)— (AgN03  dilute)Ag, 
for  which,  since  P\  =  P^  we  have  the  formula 


?r=o.ooo2T  log  —  -, 


pi  being  the  osmotic  pressure  of  ionized  Ag*  on  the  con- 
centrated side,  and  p2  that  on  the  dilute.  But,  since 
osmotic  pressure  is  proportional  to  the  concentration,  we 
may  also  use  the  formula 

n  =  0.0002  T  log  —  . 
€2 

Conductivity  experiments  show  that  the  concentration 
of  ionized  Ag'  in  a  o.oi  molar  solution  is  8.71  times  as 
great  as  that  in  one  that  is  o.ooi  molar  (not  10  times), 
hence  the  E.M.F.  at  18°  of  the  cell 

Ag(AgNO3  o.oi  molar)  —  (AgNO3  o.opi  molar)Ag 

is  TT  =  0.0002  X  291  X  log  8.71=0.054  volt,  while  direct 
measurement  shows  0.055  volt. 

Since  at  17°  we  have  TT  =  '  log  —  volts,  a  concen- 
tration ratio  of  ionized  matter  equal  to  10  would  give 


ELECTROCHEMISTRY.  209 

0.0575  voit  ror  a  monovalent  metal,  0.02875  for  a  di- 
valent one,  etc. 

Determination  of  ionization  from  electromotive-force 
measurements. — Applying  the  formula  we  have  used  for 
concentration  cell,  where  it  is  the  concentration  of  ionized 
matter  in  solution  that  varies,  it  is  very  simple  to  deter- 
mine the  concentration  of  ionized  matter  on  one  side, 
provided  that  of  the  same  kind  on  the  other  side  is  known. 
Naturally,  here,  we  can  only  apply  this  to  the  ionized 
matter  coming  from  the  electrode,  but  indirectly  from 
the  effect  that  other  kinds  of  ionized  matter  have  upon  this 
(pp.  134,  150,  1 51) 'we  can  find  the  concentration  of  the 
other  kinds.  An  example  of  the  direct  method  here,  and 
of  course  the  other  is  identical  so  far  as  the  electrical  part 
is  concerned,  for  it  simply  necessitates  after  that  the 
application  of  the  law  of  mass  action,  is  given  by  Good- 
win's determination  of  the  concentration  of  ionized  Ag* 
and  Cl'  in  a  saturated  solution  of  AgCl,  i.e.,  the  solu- 
bility of  AgCl  on  the  justified  assumption  that  the  ioniza- 
tion of  AgCl  is  practically  complete.  The  E.M.F.  at 
25°  of  the  cell 

Ag(AgN03^/io)—  KN03— (AgCl  in  KO/io)Ag 

is  0.45  volt.  Since  a  for  AgNO3m/io  is  0.82,  and  for 
KC1  m/io  is  0.85,  we  have 

0.082  0.45 

log = 3 — 3,     i.e.  c2  =  1.04X10  ~9, 

c2       0.0002X298 

where  c2  is  the  concentration  of  ionized  Ag*  in  a  saturated 
solution  of  AgCl  in  OT/io  KC1.  Since  the  concentration 
of  ionized  Cl'  in  the  KC1  is  0.085  molar,  the  solubility 
product  of  AgCl  is 

1.94  X 10  ~9Xo.o85  =  1.64  X 10 -10, 


210  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

and  its  solubility  is  \/i.64Xio~10=i.28Xio~5  moles 
per  liter  in  pure  water,  i.e.,  where  the  amount  of  ionized 
Ag*  is  equal  to  that  of  ionized  Cl'. 

In  case  we  had  had  a  solution  of  a  known  concentra- 
tion of  a  silver  salt  in  place  of  the  AgCl  in  KC1  we  could 
have  determined  the  fraction  of  ionized  Ag'  in  it.  Or, 
in  the  case  as  it  is,  we  might  dissolve  the  AgCl  in  some 
other  chloride,  and  determine,  from  the  known  solu- 
bility of  AgCl  in  pure  water  (by  conductivity,  for  ex- 
ample) and  the  law  of  mass  action,  the  concentration  of 
ionized  Cl'  in  that  salt. 

One  thing  to  be  noted  concerning  this  method  is  that 
the  smaller  the  concentration  of  ionized  Ag'  (page  203) 
on  the  left  side,  the  greater  the  E.M.F.,  and,  consequently, 
the  more  accurate  the  determination. 

Another  illustration  of  the  use  of  this  formula  in  this 
way  is  Ostwald's  determination  of  the  ionization  of  water. 
The  E.M.F.  of  the  cell  (H  in  plat.  Pt)  acid— base  (H 
in  plat.  Pt)  at  17°  is  0.081,  using  molar  acid  and  base. 
Since  a  for  the  molar  acid  is  0.8  (i.e.  H'=o.8  mole 
per  liter),  we  have 

0.8       0.81 

log  —  = ;    i.e.,  c2  =  o.8Xio~14. 

3   <2      0.0575' 

In  other  words,  the  concentration  of  ionized  H'  from 
water  in  the  presence  of  molar  base  containing  0.8  mole 
of  ionized  OH'  is  o.8Xio~14.  Hence  the  ionic  product 
for  water  is 


o.8Xo.8Xio~14,  and  since  vo.8Xo.8Xio-14  =  o.8Xio-7, 

i    liter    of    water    contains    o.8Xio~7    moles    each    of 
ionized  H'  and  ionized  OH',  which  is  the  same  value 


ELECTROCHEMIS  TRY.  2 1 1 

as  that  found  by  Kohlrausch  from  the  conductivity  of 
pure  water. 

The  processes  taking  place  in  the  cells  in  common 
use. — We  shall  now  consider,  by  aid  of  the  things  we 
have  found  above,  the  processes  which  take  place  in 
cells. 

The  Clark  cell  is  made  up  according  to  the  scheme 

Hg-Hg2S04— ZnS04-Zn. 

The  Hg2SO4  although  difficultly  soluble  goes  into  solu- 
tion to  a  slight  extent,  so  that  we  have  a  small  amount 
of  ionized  Hg2"  present  in  the  solution.  The  zinc,  owing 
to  its  high  electrolytic  solution  pressure,  goes  into  solu- 
tion and  consequently  forces  positively  charged  ionized 
matter  (Hg2")  to  give  up  its  charge.  The  zinc  is  thus  neg- 
ative from  the  loss  of  ionized  zinc,  while  the  mercury  is 
positive  owing  to  the  deposition  upon  the  electrode  of 
ionized  mercury.  The  rapid  polarization  of  the  cell 
when  short-circuited  is  due  entirely  to  the  removal  of  the 
ionized  Hg2",  but  the  value  is  restored  as  soon  as  the 
solution  again  becomes  saturated  with  the  mercury  salt, 
i.e.,  so  soon  as  the  original  concentration  of  ionized  Hg2" 
is  restored. 

The  Leclanche  cell  consists  of  a  solution  of  ammonium 
chloride,  in  which  we  have  two  electrodes,  Zn  and 
C  +  MnO2.  The  action  of  the  MnO2  is  to  prevent 
polarization,  the  processes  taking  place  without  it  and 
with  it  being  as  follows: 

In  the  cases  without  MnO2  the  Zn  with  its  high  solu- 
tion pressure  goes  into  solution,  driving  before  it  the 
other  positively  charged  matter,  i.e.,  the  ionized  NH4. 
This  ionized  matter  decomposes  on  losing  its  change, 


212  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

forming  NH3  and  H  gases.  The  bubbles  of  H  collecting 
upon  the  electrode  are  absorbed  and  so  given  off  to 
the  air,  but  as  this  process  is  slow,  the  ionized  matter 
is  prevented  from  giving  up  its  charge  and  consequently 
the  E.M.F.  decreases.  It  is  to  get  rid  of  this  action  of 
the  H  gas  that  the  MnO2  is  used.  In  contact  with  water 
we  have,  to  a  small  extent,  a  solution  of  MnO2,  which 
ionizes  according  to  the  scheme 

MnO2  +  2H2O  =  Mn  ::  +  4OH'. 

This  tetravalent  ionized  Mn::  has  the  tendency  to  go 
into  the  divalent  state  by  giving  up  two  equivalents  of 
electricity,  i.e.,  to  form  ionized  Mn".  In  consequence 
of  this  the  ionized  Mn::  with  the  ionized  NHi  is 
driven  to  the  electrode  by  the  ionized  Zn",  and,  since 
it  gives  up  two  equivalents  of  electricity  more  readily 
than  any  other  kind  of  ionized  matter  gives  up  its  entire 
charge,  the  electricity  is  given  up  by  it  without  any  sub- 
stance which  might  cause  polarization  being  deposited. 
We  have,  then,  MnCl2  (i.e.,  ionized  Mn")  formed  in  the 
solution,  and  the  process  continues  so  long  as  there  is 
solid  Zn  and  MnC>2  present. 

The   Bichromate   cell   is    arranged    according    to    the 
scheme 

Zn  -  H2SO  2—  K2Cr2O  7  -  C. 

The  action  of  the  two  substances  in  solution  forms 
chromic  acid  (H2Cr2O7).  This  ionizes  into 


to  a  considerable  extent,  and  to  a  smaller  degree  as  follows: 


ELECTROCHEM1STR  Y.  213 

This  hexavalent  ionized  Cr-::  has  the  tendency  to 
give  up  three  equivalents  of  electricity  and  to  go  into  the 
trivalent  state  (i.e.,  into  ionized  Cr").  Accordingly  the 
ionized  Zn",  which  is  forced  from  the  electrode,  drives 
before  it  the  ionized  Cr—,  which  gives  up  three  of 
its  equivalents  and  becomes  Cr"*,  remaining  as  such  in 
the  solution  as  ionized  matter  in  equilibrium  with  ionized 
SO/'  (i.e.,  as  02(804)3).  Finally,  then,  we  have  a 
solution  of  02(804)3  left  in  the  jar.  This  change  in  the 
number  of  electrical  equivalents  by  a  change  of  valence 
always  takes  place  more  readily  than  the  change  from 
the  ionic  to  the  elemental  state,  and  is  of  great  value  as  a 
means  of  preventing  polarization. 

Accumulators. — The  action  of  the  lead  accumulator 
or  storage-cell  also  depends  upon  a  change  of  valence 
Any  reversible  cell  can  be  recharged,  after  it  is  used  up, 
by  the  passage  of  a  current  through  it  in  the  direction 
opposite  to  that  in  which  it  goes  of  itself.  The  lead  cell, 
however,  is  generally  used  for  the  purpose  owing  to  its 
high  E.M.F.  Before  charging  it  consists  of  two  lead 
plates,  one  of  which  is  coated  with  litharge  (PbO),  in 
a  20%  solution  of  sulphuric  acid.  If  the  current  is 
passed  through  these  plates  (the  PbO  being  positive), 
the  PbO  is  transformed  into  PbO2,  lead  superoxide  (or 
supersulphate),  while  spongy  lead  is  deposited  upon  the 
other  electrode.  The  flow  of  current  is  now  stopped, 
and  the  cell  is  charged. 

The  PbO2  is  soluble  to  a  small  degree,  and  ionizes  as 
follows: 

PbO2  +  2H2O  =  Pb : :  +  4OH'. 

This  tetravalent,  ionized  Pb"  has  the  tendency  to  give 
up  two  of  its  electrical  equivalents  and  to  go  into  the 


214  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

divalent  form.  Since  this  is  true,  the  Pb  electrode  must 
have  the  higher  solution  pressure,  and  the  ionized  matter 
formed  from  it  will  drive  the  ionized  Pb  -  to  the  electrode, 
where  it  will  lose  two  charges  of  electricity  and  become 
divalent.  This  will  continue  as  long  as  PbO2  is  present, 
i.e.,  until  it  is  all  transformed  into  the  divalent  state, 
PbSC>4.  Other  theories,  by  Liebenow  and  others,  have 
also  been  advanced  to  explain  this  cell,  but  the  reader 
must  be  referred  elsewhere  for  them  (see  Dolezalek  and 
von  Ende,  The  Theory  of  the  Lead  Accumulator  *). 

Dolezalek  has  been  able  to  calculate  the  value  of 
such  a  cell  from  the  concentration  of  acid  and  the  vapor 
pressure  of  the  solution,  and  finds  an  excellent  agree- 
ment between  theory  and  experiment.  This  proves  the 
process,  according  to  him,  to  be  a  primary  one  such  as 
the  theory  of  Le  Blanc  and  that  of  Liebenow  would 
make  it. 

ELECTROLYSIS   AND   POLARIZATION. 

Decomposition  values. — We  must  now  consider  those 
processes  which  are  just  the  reverse  of  those  we  have 
been  considering.  In  place  of  studying  what  takes  place 
at  the  electrodes  when  a  current  of  electricity  is  pro- 
duced, we  shall  consider  the  changes  at  the  electrodes, 
and  in  the  solution,  when  the  current  is  applied  to  inert 
electrodes,  as  those  of  gold,  platinum,  carbon,  etc.  It  is 
observed  that,  when  a  current  is  passed  through  a  solu- 
tion for  a  certain  time,  using  such  electrodes,  and  then 
shut  off,  an  electromotive  force  in  the  opposite  direction 
arises;  this  is  called  the  electromotive  force  of  polarization. 

.  *  A  glance  at  this  book  will  show  the  value  in  electrochemistry  of  the 
general  physical  chemical  relations  we  have  considered,  and  will  con- 
vince the  reader  of  the  especial  impc  i  tance  of  the  law  of  mass  action. 


ELECTROCHEMISTR  Y. 


215 


Experiment  has  shown  that  every  solution  requires  a 
definite  minimum  impressed  electromotive  force  to  pro- 
duce continuous  decomposition.  Some  of  the  values 
observed  for  this  are  given  in  the  tables  below. 

DECOMPOSITION  VALUES  FOR  THE  ACIDS. 

Dextrotartaric  =  i .  62  v. 
Pyrotartaric  =  i .  5  7  v. 
Trichloracetic  =1.51  v. 
Hydrochloric  =1.31  v. 
Oxalic  =  o .  95  v. 

Hydrobromic  =  o .  94  v. 
Hydriodic  =  o .  5  2  v. 

DECOMPOSITION  VALUES  FOR  THE  BASES. 


Sulphuric              = 

.67  v. 

Nitric                    = 

.69  v. 

Phosphoric            = 

.70V. 

Monochloracetic  = 

.72V. 

Dichloracetic        = 

.66v. 

Malonic                 = 

-72V. 

Perchloric 

.65  v. 

Sodium  hydroxide          = 

Potassium  hydroxide     = 

Ammonium  hydroxide  = 

Methylamine  m/\         = 

m/2          = 

m/8         = 


69  v.  m/4 
.  67  v.  m/2 
.  74  v.  m/8 

75 
68 

74 


DECOMPOSITION  VALUES  FOR  SALTS. 


ZnS04  =2. 35v. 
Cd(NO,)2=i.98v. 
ZnBr2  =  i .  80  v. 
NiSO4  =  2 . 09  v. 

Pb(N03)2=i.52v. 


Ag(NO)3=o.7ov. 
CdSO4  =2. 03v. 
CdCLj  =i.88v. 
CoSO4  =i.94v. 
CoCLj  =  i .  78  v. 


It  will  be  observed  that  for  acids  and  bases  there  is  a 
certain  maximum  value,  which  is  reached  by  many  and 
exceeded  by  none,  and  is  equal  to  1.70  volts.  Further, 
in  all  cases  where  the  decomposition  point  is  approxi- 
mately 1.7  volts  it  is  noticed  that  the  products  of  de- 
composition are  hydrogen  and  oxygen,  and  that  those 
with  lower  values  which  usually  give  off  other  products 
also  attain  this  value  when  so  dilute  that  these  gases  are 


2i6  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

evolved.  Thus  we  find  the  following  values  for  hydro- 
chloric acid  solution,  which,  when  strong,  decompose 
into  hydrogen  and  chlorine:  2  molar  =  1.26,  1/2  molar 
=  1.34,  1/6  molar  =  1.41,  Vie  molar  =  1.62,  1/32  =  i.69. 
At  the  dilution  of  l/32  molar  hydrogqn  and  oxygen  are 
evolved. 

Le  Blanc  found  by  experiment  that  when  a  solution 
(CdSO4)  is  decomposing  steadily  the  potential  difference 
existing  between  the  cathode  (which  was  originally  of 
platinum)  is  the  same  as  that  observed  when  a  stick  of 
the  metal  which  is  deposited  is  in  contact  with  a  solution. 
Thus  a  molar  solution  of  CdSO4  is  decomposed  steadily 
at  2.03  volts,  and  when  decomposing  the  potential  dif- 
ference between  the  cathode  and  the  solution  is  +0.16 
volts,  which  is  the  same  as  that  given  by  massive  cad- 
mium in  molar  CdSC>4. 

The  process  up  to  the  decomposition-point,  then,  can 
be  readily  followed.  Originally  the  electrolytic  solution 
pressure  of  the  inert  electrode  is  zero.  The  small 
amount  of  current  which  passes  through  the  solution, 
however,  will  suffice  to  deposit  metal  upon  the  electrode, 
and  thus  increase  the  electrolytic  solution  pressure  from 
zero  to  a  definite,  small  value.  This  increase  in  P 
naturally  prevents  the  passage  of  the  current  at  the 
original  voltage.  An  increase  in  the  voltage,  then,  will 
cause  the  passage  of  current  and  the  deposition  of  more 
metal,  and  that  will  again  raise  the  electrolytic  solution 
pressure  and  prevent  the  passage  of  more  electricity  at 
this  voltage.  This  process  will  continue,  each  increase 
of  the  impressed  voltage  depositing  more  metal  and 
raising  the  electrolytic  solution  pressure.  And  only  at 
that  voltage  which  is  slightly  greater  than  the  counter 
electromotive  force  exerted  by  the  deposited  metal  in 


ELECTROCHEMISTR  Y.  217 

that  solution  will  the  decomposition  be  steady  and 
continuous. 

At  the  same  time  that  this  action  is  taking  place  at 
the  cathode  a  similar  one  proceeds  at  the  anode,  where 
the  negatively  charged  ionized  matter  is  separated  in  its 
un-  ionized  state. 

For  water  two  decomposition  values  are  observed.  The 
one  with  electrodes  of  platinized  platinum  has  the  value 
1.07  volts,  i.e.,  is  practically  the  same  as  the  electromo- 
tive force  given  by  a  gas-cell  with  such  electrodes;  the 
other,  observed  when  polished  inert  electrodes  are  em- 
ployed, being  1.68  volts.  It  will  be  seen  here  that  in 
the  first  case  we  have  a  reversible  process,  while  in  the 
second  it  is  irreversible.  It  has  been  assumed  that  water 
is  ionized  to  a  very  slight  extent  in  H*  and  O",  as  well  as 
into  H*  and  OH',  and  that  the  first,  reversible  action  is 
due  to  the  ionized  O",  the  value  1.68  volts  being  given 
when  ionized  OH'  is  separated  according  to  the  scheme 


It  is  quite  usual,  indeed,  to  designate  the  value  1.07 
as  being  due  to  O",  and  that  of  1.68  to  OH'.  Although 
we  shall  use  this  designation  (see  table,  page  219)  we 
shall  only  mean  that  O"  denotes  reversibility  and,  OH' 
irreversibility  at  the  electrode. 

Hildburgh  *  has  employed  a  reversible  electrode  with 
an  irreversible  one  in  a  device  for  rectifying  an  alter- 
nating current.  He  used  a  large  piece  of  platinized 
platinum  for  one  electrode,  the  other  being  a  small  point 
of  polished  platinum,  while  the  electrolyte  is  a  solution 
of  sulphuric  acid  which  covers  one-half  of  the  large 
electrode.  Before  being  stoppered  the  bottle,  containing 

*  Jour.  Am.  Chem.  Soc.,  22,  300,  1900.         ; 


218  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

the  sulphuric  acid  solution  and  the  electrodes,  is  filled  with 
hydrogen  gas  at  a  certain  pressure  and  sealed.  When 
the  point  is  the  cathode,  i.e.  the  large  platinized  elec- 
trode is  the  anode,  it  is  observed  that  we  get  a  continu- 
ous decomposition  of  water  at  about  i.i  volts,  hydrogen 
being  evolved  from  the  point  and  oxygen  absorbed  in  the 
platinum  black.  When  the  point  is  the  anode,  however, 
we  get  bubbles  of  oxygen  first  at  1.68  volts,  the  hydrogen 
being  absorbed  in  the  platinum  black.  In  this  way,  by 
taking  enough  cells  in  series,  one  can  rectify  an  alternat- 
ing current,  i.e.,  transform  it  into  a  series  of  impulses  all 
in  the  one  direction. 

Primary  and  secondary  decomposition  of  water. — 
The  electromotive  force  of  decomposition  of  a  substance 
giving  off  hydrogen  and  nitrogen  is  dependent  upon 
the  concentration  of  the  ionized  H*  and  ionized  OH', 
but  independent  of  the  nature  of  the  electrolyte.  The 
decomposition  value  is  thus  the  same  for  acids  and  for 
bases,  so  long  as  only  hydrogen  and  oxygen  are  separated. 
Since  by  the  law  of  mass  action  the  product  of  the  con- 
centration of  the  ionized  H'  and  the  ionized  OH'  must 
always  be  the  same  in  any  water  solution,  it  follows  that 
for  all  electrolytes,  since  the  electromotive  force  of  the 
cell  is  the  sum  of  the  differences  of  potential  at  the  two 
electrodes,  the  minimum  value  must  be  the  same  for  all 
substances  giving  off  oxygen  and  hydrogen.  With  the 
exception,  then,  of  the  solutions  of  metallic  salts  which 
are  decomposed  by  hydrogen,  and  the  chlorides,  bromides, 
and  iodides  which  are  decomposed  by  oxygen,  the  ionized 
products  of  water  are  the  only  factors  in  the  decompo- 
sition of  solutions,  and  not  those  of  the  dissolved  salt. 
Excluding  these  solutions,  then,  we  may  say  that  all 
solutions  when  electrolyzed  show  primary  decomposition 


ELECTROCHEMISTRY.  219 

of  water.  The  current  is  conducted  through  the  solu- 
tion by  all  the  kinds  of  ionized  matter  which  are  present. 
At  the  electrodes,  however,  that  process  takes  place  which 
involves  the  expenditure  of  the  smallest  amount  of  work 
and  that  is  the  separation  of  hydrogen  and  oxygen. 
Thus  we  see  in  all  cases  that  there  must  be  an  accumula- 
tion of  the  various  kinds  of  ionized  matter  around  the 
electrode,  but  that  only  that  kind  is  separated  which  does 
so  most  easily.  Naturally,  if  the  amount  of  current  con- 
ducted through  the  liquid  is  so  great  that  hydrogen  and 
oxygen  cannot  be  separated  with  the  least  work  (owing  to 
the  small  concentration  of  ionized  H*  and  ionized  OH') 
some  other  material  may  be  separated  instead,  which  will 
then  decompose  the  water.  But  for  small  currents  it 
is  undoubtedly  true  that  the  decomposition  of  water  is 
primary,  and  not  secondary. 

In  the  table  below  are  given  the  values  necessary  for 
the  separation  of  the  various  kinds  of  ionized  matter,  on 
the  assumption  that  that  for  ionized  H*  is  zero.  The 
values  are  for  molar  solutions. 


Ag-  =-0.78 
Cu"=-o.34 
H'    =+o.o 
Pb"=+o.i7 
Cd"=+o.38 


r      =0.52 

Br'  =0.94 

O"  =  i.  08  (in  acid) 

Cl'  =1.31 

OH'  =1.68  (in  acid) 

OH'  =0.88  (in  base) 

SO4"  =1.9 

HS04'=2.6 


The  values  of  O"  and  OH'  are  true  in  the  presence  of  a 
molar  solution  of  ionized  H'.  If  we  have  H'  and  OH' 
in  a  base,  the  above  value  of  H"  becomes  0.8  and  the 
value  of  OH'  and  O"  is  decreased  by  0.8. 


CHAPTER  VIII. 

PROBLEMS.* 
GASES. 

1.  An  open  vessel  is  heated  to  819°  C.     What  por- 
tion of  the  air  which  the  vessel  contained  at  o°  remains 
in  it?  Ans.  0.25. 

2.  An  open  vessel  is  heated  until  one-half  of  the  gas 
contained  at  15°  is  driven  out.     What  is  the  temperature 
of  the  vessel?  Ans.  303°  C. 

3.  A  volume  of  gas,  measured  at  15°,  is  50  c.c.     At 
what  temperature  would  its  volume  become  44  c.c.  ? 

Ans.   -i9°.6  C. 

4.  A  volume  of  gas  at  766  mm.  pressure  is  137  c.c. 
What  would  it  be  at  757  mm.?  Ans.  138.7  c.c. 

5.  What  volume  does  i   mole  of  gas  occupy  at  50°, 
the  pressure  being  760  mm.  ?    At  100°,  p  being  900  mm.  ? 

Ans.  ?;soo=26.5,  at  *>IOO°  =  25.8  liters. 

6.  A  volume  of  air  in  a  bell  jar  over  water  measures 
975  c.c.     The  water  in  the  jar  is  68  mm.  above  the  water 
in  the  trough,  and  the  barometer  stands  at  756  mm. 
What  would  the  volume  be  if  exposed  to  standard  pres- 
sure, the  specific  gravity  of  Hg  being  13.6?  Ans.  963.4. 

7.  At   14°  C.    and    742    mm.    pressure    a   volume    of 
gas  measures  18  c.c.      What   will   be  its  volume  at  o° 
and  760  mm.  pressure?  Ans.  16.72. 

*  for  further  problems,  see  "  Elements,"  pp.  453-485. 

220 


PROBLEMS.  221 

8.  A  volume  of  H  at  a  temperature  of  15°  measures 
2.7  liters  with  the  barometer  at  752  mm.     What  would 
have  been  its  volume  had  the  temperature  been  9°  and 
the  pressure  762  mm.  ?  Ans.  2.6  liters. 

9.  What  volume  is  occupied  by  44  grams  of  oxygen  at 
70  cm.  Hg  pressure  and  35°  C.  ?  Ans.  37.7  liters. 

10.  J  mole  of  H,  J  mole  of  O,  and  $  mole  of  N  are 
mixed  in  a  volume  of  10  liters  at  o°C.  ,  What  are  the 
partial  pressures  of  H,  O,  and  N? 

Ans.  pH  =  1156.96,  p0=  1156.96,  and  pN=  771.65  grams 
per  sq.  cm. 

11.  What  would  these  pressures   (10)  be  in    atmos- 
pheres at  10°  C.  ? 

Ans.  pn=i.i64,  p0  =  i.i64,  and  ^=0.774- 

12.  i  liter  of  N  weighs  1.2579  grams  at  o°  and  760 
mm.     Calculate  the  specific  gas  constant,  r. 

Ans.  3007  grams  per  sq.  cm. 

13.  The  specific  gas  constant,  r,  for  N  was  found  above 
(12).     What  is  it  for  H  ?    The  combining  weight  of  N  is 
14.04,  and  of  H  is  i  .008.     Ans.  39,080  grams  per  sq.  cm. 

14.  How  much  will  100  liters  of  chlorine  at  74  cm. 
Hg  pressure  and  30° C.   weigh?    Ans.  278.7  grams. 

15.  A  solid  gives  off   a  gas  which   is  dissociated  to 
41%,  into  two  products.     What  is  the  work  done,  in 
calories,  gram-centimeters,  and  liter-atmospheres,  when 
i  mole  of  solid  goes  into  the  gaseous  state,  the  tempera- 
ture of  dissociation  being  55°  C.  ? 

Ans.  925  cals.,  39,410,000  gr.-cm.,  37.96  L.  A. 

1 6.  How  much  work  will  be  done  by  i  kg.  of  CO 2 
when  heated  200°?  Ans.  373.1  L.  A.,  9088  cals. 

17.  H  is  at  the  partial  pressure  of  2.136  atmospheres 
in  a  space  of  10  liters.     How  many  moles  per  liter  are 
there,  the  temperature  being  o°?  Ans.  c= 0.0954. 


222  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

1 8.  Starting  with  i  mole  of  A  in  22.4  liters  (at  o°,  760 
mm.  of  Hg),  assume  the  dissociation  according  to  the 
scheme  A  =2B  +  $D  (where  A,  B,  and  D  represent  moles) 
to  be  23%.     What  will  be  the  final  volume  where  pres- 
sure and  temperature  remain  unchanged  ?     Ans.  43  liters. 

19.  What  are  the  final  concentrations  of  A,  5,  and  D 
in  the  above? 

Ans.  A  =0.0179,  5=0.0107,  and  D  =  0.01604  rnole  per 
liter. 

20.  The  formula  weights  of  the  above  are  M A  =  1 70, 
MB  =  25,  and  ^#  =  40.     How  many  grams  per  liter  are 
there  of  each  at  equilibrium? 

Ans.  ^4=3.04,  5=0.268,  and  12=0.642. 

21.  Assume  17  grams  of  A  (If  =  170)  in  2.24  liters  (o°, 
760  mm.   Hg).      Find  concentrations,  partial  pressures 
and  grams  per  liter  of  A ,  5,  and  D  where  the  dissociation 
of  A  is  20%  (M#  =  25,  M£>=4o),  and  the  volume  and 
temperature  remain  constant.     What  is  the  total  pres- 
sure of  the  system? 

Ans.  ;!  =0.036,  5=o.oi8,  Z>= 0.027  mole  per  liter. 
^4=6.o6,     5  =  0.447,  Z>  =  i.o7  gr.  per  liter. 
^4=o.8,       5  =  0.4,      D  =  0.6  atmosphere. 
Total  pressure  =  i. 80  atmospheres. 

22.  When  heated,  PC15  dissociates  into  PC13  and  C12. 
The  formula  weight  of  PC15  is   208.28.     At   182°  the 
density  is  73.5,  and  at  230°  it  is  62.     Find  the  degree 
of  dissociation  at  182°  and  230°. 

23.  Sulphur  in  the  form  S8  dissociates  under  certain 
conditions  into  the  form  S2.     If  this  dissociation  were 
complete,  what  would  be  the  density  of  the  gas  in  the 
form  S2?  Ans.  32. 

24.  The  specific  heat  under  constant  pressure  for  helium 


PROBLEMS.  223 

is  1.25;   the  formula  weight  is  4.     How  many  combining 
weights  are  there  in  the  formula  weight?        Ans.  i. 

25.  What  is  the  specific  heat  at  constant  volume,  i.e., 
cv?     i  gram  of  the  gas  at  o°  and  760  mm.  occupies  509 
c.c.,  and  cp=o.2i.  Ans.  0.164. 

26.  The  specific  heat  at  constant  volume  of  a  sub- 
stance is  0.075;    its  formula  weight  is  40.     How  many 
combining  weights  are  there  to  one  formula  weight? 

Ans.  i. 

27.  The  specific  heat,  cv,  of  CC>2  is  0.2094;    what  is 
the  ratio  of  that  for  constant  pressure  to  that  at  con- 

stant volume?  Ans.  ~  =  i.22. 


SOLUTIONS. 

28.  What  is  the  osmotic  pressure  of  a  i%  solution  of 
glucose  (M  =  iSo)  at  o°C.? 

Ans.  94.6  cm.  H;  obs.  =94  cm. 

29.  The  osmotic  pressure  of  a  solution  of  cane-sugar 
at  o°  is  49.3   cm.   of  Hg.     What  percentage  of  sugar 
(M  =  342)  is  contained  in  it  ?   Ans.  0.99%  ;  obs.  =  1.0%. 

30.  The  osmotic  pressure  of  a  sugar  solution  at  32°  CD- 
is  54.4  mm.     What  is  it  at  i4°.2  ?          Ans.  51.2  mm. 

31.  The  osmotic  pressure  of  solution   containing   10 
grams  of  sugar  to  a  certain  volume  is  200  mm.     What 
is  that  for  the  same  volume  containing  13.5  grams? 

Ans.  270  mm. 

32.  10.442  grams  aniline  in  100  grams  of  ether  give  a 
vapor  pressure  of  210.8  mm.     Ether  alone  (M  =  74)  gives 
229.6.     Find  the  formula  weight  of  aniline  in  ether. 

Ans.  87. 

33.  Find  osmotic  pressure  at  o°  of  aniline  in  (32)  in  at- 


224  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

mospheres  and  gram  per  square  centimeter,     (s  for  ether 
is  0.737.)     Ans.  19.82  atmos.  or  20,460  gr.  per  sq.  cm. 

34.  The  osmotic  pressure  of  a  substance  in  water  solu- 
tion is  100  cm.  at  o°  C.      Find  the  vapor  pressure  of 
the  solution;    that  of  water  at  o°  is  4.57  mm. 

Ans.  4.56  mm. 

35.  What  is  the  work,  in  gr.-cms.  and  calories,  neces- 
sary to  separate  200  grams  of  a  substance  (M  =  6o)  from 
the  solvent  at  20°  C.  ?     10  grams  of  substance  to  the  liter 
of  solvent.  Ans.  1953  cals. ;  83,200,000  gr.-cm. 

36.  The  increase  in  the  boiling-point  of  54.65  grams  of 
CS2  caused  by  the  addition  of  1.4475  grams  of  P  is  o°.486. 
What  is  the  formula  weight  of  P  in  CS2?    Ans.  129.2. 

What  is  the  formula,  the  combining  weight  being  31  ? 

37.  Calculate  the   increase    in    boiling-point  of  ether 
when  to  100  grams  we  add  a  mole  of  a  substance.     The 
boiling-point  of  ether  is  34°.97,  the  latent  heat  of  evapo- 
ration is  88.39.  Ans.  21°. 5. 

38.  The   molecular   increase    of   the    boiling-point   of 
H2O,  as  caused  by  the  addition  of  i  mole  of  substance 
to  100  grams,  is  5°.2.     Find  heat  of  evaporation  of  H2O. 

Ans.  535.1  cals. 

39.  In  (36)  find  the  osmotic  pressure  at  46°  of  P  in 
the  CS2  solution.     (sCS2  =  1.2224.)        Ans.  6.84  atmos. 

40.  Find  the  vapor  pressure  in  (36)  of  P  in  CS2  solu- 
tion at  o°;  the  vapor  pressure  of  CS2  at  o°  is  127.91  mm. 

Ans.  125.9  mm- 

41.  10  grams  of  a  substance  in  100  grams  of  a  solvent 
increase  the  boiling-point  by  o°.87.     The  formula  weight 
of  the  substance  is  60.     Find  the  molecular  increase  of 
the  boiling-point.  Ans.,  5.22. 

42.  0.284  gram  of  the  oxime  (CHs)2CNOH  causes  a 
decrease  of  o°.i55  in  the  freezing-point  of  100  grams  of 


PROBLEMS.  225 

glacial   acetic   acid,     k   for   acetic    acid   is    38.8.     Find 
the  formula  weight  of  the  oxime  in  acetic  acid.     Ans.  71. 

43.  The  ionization  of  a  molar  solution  is  80%,  two 
kinds  of  ionized  matter  being  formed.     What  will  the 
depression    of    the    freezing-point    be,    water    (£  =  18.9) 
being  the  solvent?  Ans.  ^.4. 

44.  The  molecular  depression  of  an  aqueous  solution 
containing  an  ionized  substance  is  22°.     Find  the  degree 
of  ionization  of  the  substance  in  that  volume. 

Ans.  a  =  16.3%. 

45.  In  (42)  find  the  osmotic  pressure  of  the  oxime  in 
glacial  acetic  acid  at  17°.     (Sp.  gr.  of  acetic  acid  =  1.056.) 

Ans.  i  atmos. 

46.  What  is  the  relation  between  the  osmotic  pres- 
sures of  o.oi  mole  of  substance  in  1000  grams  of  water 
and  looo  grams  of  ether  (sp.  gr.  =0.7370),  assuming  the 
same  formula  weight  of  the  solute  in  each? 

Ans.  Pe  =  o.j^oPw. 

47.  In   (42)  find  the  vapor  pressure  of  the  solution 
at  40°  C.,  the  vapor  pressure  of  glacial  acetic  acid  at 
40°  being  34.77  mm.  Ans.  34.69  mm. 

48.  A   o.i    molar   solution    of   acetic   acid   in   water 
freezes  o°.i927   lower  than  H2O.     Find  the  degree  of 
ionization  of  the  acetic  acid.  Ans.  a  =  2%. 

49.  A  0.15    molar    solution  of    succinic    acid  freezes 
o°.2864  lower  than  H2O.     Find   the  ionization  of  the 
acid.  Ans!  a  =  i%. 

50.  What  is  the  heat  of  formation  of  a  very  dilute 
solution  of  magnesium  chloride?     (See  text.) 

Ans.  1875  cals. 


226  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 


CHEMICAL  MECHANICS. 

51.  In  the  volume  of  i  liter  there  are  0.14  mole  of  hy- 
drogen and  0.081  mole  of  iodine.     At  the  temperature  of 

440°  C.  K  =  ^j2  =0.02.     Find  the  amount  of  hydriodic 

acid  formed.  Ans.  0.14855  mole. 

52.  The  initial  pressure  of  I  is  38.2  cm.,  the  fraction 
uniting  with  H  is  0.8.     What  was  the  original  pressure 
of  H,  2  =  440°?     (K  =  o.02.)  Ans.  40.35  cm. 

53.  At  440°  in  50  liters  we  have  a  mixture  of  2.74  moles 
of  HI,  0.5  mole  of  H,  and  0.3011  mole  of  I.  (^=0.02.) 
In  which  direction  and  to  what  extent  will  the  reaction  go  ? 

54.  At  440°  (K=o.o2)  5.30  c.c.  of  H  are  mixed  with 
7.94  c.c.  of  I.     How  much  HI  will  be  formed? 

Ans.  9.475  c.c.;  observed,  9.52  c.c. 

55.  6.63  moles  of  amylene  with  i  mole  of  acid  shows 
that  0.838  mole  is  formed  in  the  total  volume  of  894 
liters.     How  much  will  be  formed  when  we  start  with 
4.48  moles  of  amylene  and  i  of  acid  in  the  volume  of 
683  liters?  Ans.  0.811 1  mole. 

56.  The  coefficient  of  distribution  of  acetic  acid  between 

water  and  benzene  is  as  - — -  and  — —  at  two  dilutions. 

0.043          0.071 

What  is  the  formula  weight  of  acetic  acid  in  benzene? 
In  water  it  is  60.  Ans.  2.02X60. 

57.  For    the    reaction— solid    NH4HS=H2S  +  NH3— 
K  =  62,400    (for   pressures   in   mm.)    at    25°.!  C.      In  a 
vacuum  at  25°.!  we  introduce  NH3  and  H2S  until  we 
have  a  partial  pressure  of  the  former  of  300  mm.,  and 
of  the  latter  of  594  mm.     Then  the  reaction  is  allowed 
to  take  place.     How  much  does  each  gas  lose  in  pres- 


PROBLEMS.  227 

sure?     (Here   the   pressure   of   gaseous    NH^HS  is   so 
small  as  to  be  negligible.)  Ans.  157.2  mm. 

58.  At    i8°.4   i    mole   of   BaSO4   dissolves  in   50,055 
liters;  at  37°. 7  in  31,282  liters.     On  the  justified  assump- 
tion that  the  BaSCU  is  completely  ionized,  calculate  the 
heat  of  ionization  per  mole.  Ans.   —8511  cals. 

EQUILIBRIUM   IN   ELECTROLYTES. 

59.  To  i  liter  of  a  molar  solution  of  a  monobasic  acid 
(K= 0.000018),    a   binary   salt,   with   ionized   matter  in 
common,   having   an   ionization   in   that   dilution   equal 
to  100%,  is  added.     How  much  (in  mo  les)  in  the  dry 
state  must  be  dissolved  in  the  acid  solution  to  decrease 
the  concentration  of  ionized  H*  to  o.i   of  its  previous 
value?  Ans.  0.04211  mole. 

60.  A  small  amount  of  base  is  mixed  with  an  excess 
of  a  solution  containing  an  equal  number  of  formula 
weights  of  acetic  and  lactic  acids.     In  what  proportion 
will  the  corresponding  salts  be  formed? 

Ans.  Lactate  :  acetate:  10.0117:  0.00424. 

61.  PbI2  is  soluble  to  0.00158  mole  per  liter  at  25°.2, 
and  ionizes  completely  (practically)  into  Pb"  and  2!'. 
What  is  its  solubility  in  presence  of  a  o.i  molar  solution 
of  ionized  I'  from  another  salt? 

Ans.  i . 5 8 4- 1 o6  moles  per  liter. 

62.  The  solubility  product  of  the  substance  AC2  is 
0.00621.     What  is  the  concentration  of  ionized  A"  and 
C'  when  the  ionization  is  complete  into  A"  and  2C'? 

Ans.  0.1157  mole  per  liter  of  A",  and  0.2314  of  C'. 

63.  MCN    (solubility    is   0.02    and    is    ionized    com- 
pletely) is  hydrolytically  dissociated  in  solution.     K  for 
HCN  is  i3Xio-10,  and  K  for  H2O  (25°)  is  (1.09X10-7)2. 


228  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

Find  the  amount  of  ionized  M*  from  another  salt  which 
must  be  present  to  prevent  hydrolytic  dissociation. 

Ans.  0.02162  mole  per  liter. 

64.  At   the  dilution   of  32   liters   a  binary  substance 
is  0.9%  hydrolyzed.     What  is  the  percentage  of  hydroly- 
sis at  the  same  temperature  when  the  dilution  is  100 
liters?  Ans.  1.584%. 

65.  The  constant  of  hydrolytic  dissociation  at  100°  for 

NH4C1  is  337X10-10(1.6.,  *=(7-     What    is 


the  ionization  constant  of  NH^OH?     (sH2o  at   100°  is 
(8.5XIQ-7)2.)  Ans.  214X10-7. 

66.  AgCl,    AgBr,    and    Agl    are    dissolved    together. 
What  are  the  concentrations  of  ionized  Ag*,   Bi^,   Cl', 
and    T?    The    solubilities    are    i.25Xio~5,    86Xio~8, 
and  0.97  Xio~8  respectively. 

67.  Bromisocinnamic  acid  at  25°  is  soluble  to  0.0176 
mole  per  liter,  and  is    ionized  to  1.76%  into  H*  and 
a   negatively   charged   radical.     What   is   the   solubility 
product  of  the  acid?  Ans.  9.6Xio~8. 

What  amount  of  the  acid  must  always  remain  in  solu- 
tion at  this  temperature,  even  in  the  presence  of  an  in- 
finite amount  of  ionized  H*  from  another  acid? 

Ans.  0.0173  m°le  Per  liter. 

What  is  the  solubility  of  the  acid  in  the  presence 
of  a  o.oo  i  molar  solution  of  ionized  H*  from  another 
acid? 

A  ns.  Solubility  =  o.oi  73  4-  0.0000883  mole  per  liter. 

68.  Find  the  heat  of  neutralization  of  i  mole  of  acetic 
acid  (in  200  moles  (3600  gr.)  of  H^O)  with  i  mole  of 
sodium  hydrate  (in  200  moles  of  H2O)  at  35°.     (a  for 
acetic  acid  is  0.009,  its  heat  of  ionization  is  —386  cals.; 
a  for  NaOH  is  0.861,  its  heat  is  —1292  cals.,  and  a  for 


PROBLEMS.  229 

sodium  acetate  is  0.742,  its  heat  being  -391  cals.    The 
heat  of  ionization  of  water  at  35°  is  12,632  cals.) 

Ans.  13,093  cals. 

ELECTROCHEMISTRY. 

69.  An   aqueous   solution   of   CuSC>4   is   electrolyzed 
until    0.2955    gram    of    Cu    is    deposited,    using    inert 
electrodes.     The    solution    at    the  .cathode    before    the 
passage  of  the  current  gave  2.2762   grams  of  Cu,  and 
after  the  passage  2.0650  grams.     Find  the  mobility  of 
ionized  Cu"  and  of  ionized  SO4". 

Ans.  tf 0^=0.285,  #scy'=o.7i5. 

70.  A  0.02  molar  solution  of  KC1  (£=0.002397)  gives 
in  a  certain  cell  a  resistance  of  150  ohms.     What  is  the 
factor  that  will  transform  conductivity  results  determined 
in   this   cell   into   specific   conductivities?    Ans.  0.36. 

71.  The    equivalent    conductivity    of    a    solution    of 
Na2SO4  in  256  liters  at  25°  is  141.9.     What  is  it  at  in- 
finite dilution  ?  Ans.  153.9. 

72.  The  conductivity  of  a  solution  of  AgCl  saturated 
at  i8°is  2.4Xio~6;  that  of  the  water  used  is  i.i6Xio~6. 
Find  the  solubility  of  AgCl  on  the  justified  assumption 
that  it  is  completely  ionized.     (^0oKa=I3I-2>  ^<»AgNOa= 
116.5,  and  AxKN03  =  126.1.) 

Ans.  i.o2Xio~5  moles  per  liter. 

73.  Find  the  heat  of  amalgamation  of  cadmium  at 
o°.      (x  for  a  ceU  made  up  of  a  i%  cadmium  amalgam 
and  mercury  in  a  solution  of  CdSC>4  is  0.06836  volt  at 
o°  and  0.0735  at  24°-45-)  Ans.  #  =  510  cal.  per  mole  of  Cd. 

74.  Zn  in  a  molar  solution  of  ZnSC>4  gives  a  difference 

An 
of  potential  of  0.51  volt  and  7™  =  —0.00076.     What  is 

the  heat  of  ionization  of  Zn  at  17°?    Ans.  33,740  cals. 


230  PHYSICAL  CHEMISTRY  FOR  ELECTRICAL  ENGINEERS. 

75.  A   cell   with   electrodes  of  the  same  monovalent 
metal  gives  an  E.M.F.  at  17°  of  0.35  volt.     The  con- 
centration of  ionized  metal  at  the  positive  electrode  is 
0.02  mole  per  liter.     What  is  its  concentration  at  the  other 
electrode?     (The   two   solutions   are   connected   with   a 
siphon  to  prevent  diffusion.) 

Ans.  i. 637  Xio~8  moles  per  liter. 

76.  What  would  be  the  E.M.F.  of  a  concentration  cell 
with  electrodes  of  a  divalent  metal,  the  concentration  of 
ionized  metal  being  0.02  and  1.637  Xio~6  moles  per  liter? 

Ans.  0.175  v0^- 

77.  In  a  hydrogen-gas  cell  (platinized  platinum  elec- 
trodes, one-half  in  solution,  one-half  in  gas)  we  have 
acetic  acid  on  one  side  and  propionic  on  the  other,  the 
concentration  being  identical,  i.e.  molar.     What  is  the 
E.M.F.  of  the  cell  at  17°?   Which  is  the  positive  electrode  ? 

Ans.  Acetic  acid  positive,  71  =  0.00369  volt. 

78.  What  is  the  relation  of  the  electrolytic  solution 
pressure  of  Zn  to  that  of  Cu?     (The  E.M.F.  of  Cu  in 
CuSO*  against  Zn  in  ZnSC>4  is  1.06  volts  when  the 
concentration  of  ionized  Zn"  is  equal  to  that  of  ionized 

^  Ans.       * 


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Winthrop's  Abridgment  of  Military  Law 1 2mo,  2  50 

Woodhull's  Notes  on  Military  Hygiene i6mo,  i  50 

Young's  Simple  Elements  of  Navigation i6mo,  morocco,  i  oo 

Second  Edition,  Enlarged  and  Revised i6mo,  morocco,  2  oo 

ASSAYING. 

Fletcher's  Practical  Instructions  in  Quantitative  Assaying  with  the  Blowpipe. 

i2mo,  morocco,  i  50 

Furman's  Manual  of  Practical  Assaying 8vo,  3  oo 

Lodge's  Notes  on  Assaying  and  Metallurgical  Laboratory  Experiments ....  8vo,  3  oo 

Miller's  T'anual  of  Assaying I2mo,  i  oo 

O'Driscoll's  Notes  on  the  Treatment  of  Gold  Ores 8vo,  2  oo 

Ricketts  and  Miller's  Notes  on  Assaying 8vo,  3  oo 

Ulke's  Modern  Electrolytic  Copper  Refining 8vo,  3  oo 

Wilson's  Cyanide  Processes I2mo,  i  50 

Chlorination  Process i2mo,  i  50 

ASTRONOMY. 

Comstock's  Field  Astronomy  for  Engineers 8vo,  2  50 

Craig's  Azimuth 4to,  3  50 

Doolittle's  Treatise  on  Practical  Astronomy 8vo,  4  oo 

Gore's  Elements  of  Geodesy 8vo,  2  50 

Hayford's  Text-book  of  Geodetic  Astronomy; 8vo,  3  oo 

Merriman's  Elements  of  Precise  Surveying  and  Geodesy 8vo,  2  50 

*  Michie  and  Harlow's  Practical  Astronomy 8vo,  3  oo 

*  White's  Elements  of  Theoretical  and  Descriptive  Astronomy i2mo,  2  oo 

BOTANY. 

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i6mo,  morocco,  i  25 

Thome'  and  Bennett's  Structural  and  Physiological  Botany : i6mo,  2  25 

Westermaier's  Compendium  of  General  Botany.     (Schneider.) 8vo,  2  oo 

CHEMISTRY. 

Adriance's  Laboratory  Calculations  and  Specific  Gravity  Tables I2mo,  i  25 

Allen's  Tables  for  Iron  Analysis 8vo,  3  oo 

Arnold's  Compendium  of  Chemistry.     (Mandel.) Small  8vo,  3  50 

Austen's  Notes  for  Chemical  Students i2mo,  i  50 

Bernadou's  Smokeless  Powder. — Nitro-celhilose,  and  Theory  of  the  Cellulose 

Molecule i2mo,  2  50 

Bolton's  Quantitative  Analysis 8vo,  i  50 

*  Browning's  Introduction  to  the  Rarer  Elements. 8vo,  i  50 

Brush  and  Penfield's  Manual  of  Determinative  Mineralogy. 8vo,  4  oo 

Classen's  Quantitative  Chemical  Analysis  by  Electrolysis.    (Boltwood. ) .  .  8vo,  3  oo 

Conn's  Indicators  and  Test-papers i2mo,  2  oo 

Tests  and  Reagents ...    8vo,  3  oo 

Crafts's  Short  Course  in  Qualitative  Chemical  Analysis.  (Schaeffer.). .  .i2mo,  i  50 
Dolezalek's  Theory  of  the  Lead  Accumulator  (Storage  Battery).        (Von 

Ende.) I2mo,  2  50 

Drechsel's  Chemical  Reactions.     (Merrill.) I2mo,  i  25 

Duhem's  Thermodynamics  and  Chemistry.     (Burgess.) 8vo,  4  oo 

Eissler's  Modern  High  Explosives 8vo,  4  oo 

Eff rent's  Enzymes  and  their  Applications.     (Prescott) 8vo,  3  oo 

Erdxnann's  Introduction  to  Chemical  Preparations.     i.Dunlap.) I2mo.  i  25 

3 


Fletcher's  Practical  instructions  in  Quantitative  Assaying  with  the  Blowpipe. 

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Fowler's  Sewage  Works  Analyses i2mo,  2  oo 

Fresenius's  Manual  of  Qualitative  Chemical  Analysis.     (Wells.) 8vo,  5  oo 

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2  vols 8vo,  12  50 

Fuertes's  Water  and  Public  Health i2mo,  i  50 

Furman's  Manual  of  Practical  Assaying 8vo,  3  oo 

*  Getman's  Exercises  in  Physical  Chemistry I2mo,  2  oo 

Gill's  Gas  and  Fuel  Analysis  for  Engineers i2mo,  i   25 

Grotenfelt's  Principles  of  Modern  Dairy  Practice.     (Woll.) i2mo,  2  oo 

Hammarsten's  Text-book  of  Physiological  Chemistry.     (Mandel.) .8vo,  4  oo 

Helm's  Principles  of  Mathematical  Chemistry.     (Morgan.) i2mo,  i  50 

Bering's  Ready  Reference  Tables  (Conversion  Factors) i6mo,  morocco,  2  50 

Hind's  Inorganic  Chemistry 8vo,  3  oo 

*  Laboratory  Manual  for  Students i2tno,  75 

Holleman's  Text-book  of  Inorganic  Chemistry.     (Cooper.) 8vo,  2  50 

Text-book  of  Organic  Chemistry.     (Walker  and  Mott.) 8vo,  2  50 

*  Laboratory  Manual  of  Organic  Chemistry.     (Walker.) i2mo,  i  oo 

Hopkins's  Oil-chemists'  Handbook 8vo,  3  oo 

Jackson's  Directions  for  Laboratory  Work  in  Physiological  Chemistry.  .8vo,  i  25 

Keep's  Cast  Iron 8vo,  2  50 

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Chemistry.  (Tingle.) i2mo,  i  oo 

Leach's  The  Inspection  and  Analysis  of  Food  with  Special  Reference  to  State 

Control 8vo,  7  50 

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Lodge's  Notes  on  Assaying  and  Metallurgical  Laboratory  Experiments.  ..  .8vo,  3  oo 

Lunge's  Techno-chemical  Analysis.  (Cohn.) i2mo,  i  oo 

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Mason's  Water-supply.     (Considered  Principally  from  a  Sanitary  Standpoint.) 

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Miller's  Manual  of  Assaying i2mo,  oo 

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Elements  of  Physical  Chemistry i2mo,  oo 

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Mulliken's  General  Method  for  the  Identification  of  Pure  Organic  Compounds. 

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Pictet's  The  Alkaloids  and  their  Chemical  Constitution.     (Biddle.) 8vo,  5  oo 

Pinner's  Introduction  to  Organic  Chemistry.     (Austen.) i2mo,  i  50 

Poole's  Calorific  Power  of  Fuels 8vo,  3  oo 

Prescott  and  Winslow's  Elements  of  Water  Bacteriology,  with  Special  Refer- 
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Richards  and  Woodman's  Air,  Water,  and  Food  from  a  Sanitary  Standpoint  8vo,  2  oo 

Richards's  Cost  of  Living  as  Modified  by  Sanitary  Science i2mo,  i  oo 

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Ricketts  and  Russell's  Skeleton  Notes  upon  inorganic  Chemistry.     (Part  I. 

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Ricketts  and  Miller's  Notes  on  Assaying 8vo,  3  oo 

Rideal's  Sewage  and  the  Bacterial  Purificat'on  of  Sewage 8vo,  3  50 

Disinfection  and  the  Preservation  of  Food 8vo,  4  oo 

Rigg's  Elementary  Manual  for  the  Chemical  Laboratory 8vo,  i  25 

Rostoski's  Serum  Diagnosis.  (Bolduan.) i2mo,  i  oo 

Ruddiman's  Incompatibilities  in  Prescriptions 8vo,  2  oo 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish 8vo,  3  oo 

SaJkowski's  Physiological  and  Pathological  Chemistry.  (Orndorff.) 8vo,  250 

Schimpf's  Text-book  of  Volumetric  Analysis I2mo,  2  50 

Essentials  of  Volumetric  Analysis i2mo,  i  25 

Spencer's  Handbook  for  Chemists  of  Beet-sugar  Houses i6mo,  morocco,  3  oo 

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Stockbridge's  Rocks  and  Soils 8vo,  2  50 

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Treadwell's  Qualitative  Analysis.     (HalL) 8vo.  3  oo 

Quantitative  Analysis.     (HalL) 8vo,  4  oo 

Turneaure  and  Russell's  Public  Water-supplies 8vo,  5  oo 

Van  Deventer's  Bhysical  Chemistry  for  Beginners.     (Boltwood.) i2mo,  i  50 

*  Walke's  Lectures  on  Explosives 8vo,  4  oo 

Washington's  Manual  of  the  Chemical  Analysis  of  Rocks 8vo,  2  oo 

Wassermann's  Immune  Sera:  Heemolysins,  Cytotoxius,  and  Precipitins.    (Bol- 
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Well's  Laboratory  Guide  in  Qualitative  Chemical  Analysis 8vo,  i  50 

Short  Course  in  Inorganic  Qualitative  Chemical  Analysis  for  Engineering 

Students i2mo,  i  50 

Text-book  of  Chemical  Arithmetic.     (In  press.) 

Whipple's  Microscopy  of  Drinking-water 8vo,  3  50 

Wilson's  Cyanide  Processes i2mo,  i  50 

Chlorination  Process i2mo,  i  50 

Wulling's    Elementary    Course    in  Inorganic,  Pharmaceutical,  and  Medical 

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CIVIL  ENGINEERING. 

BRIDGES   AND    ROOFS.       HYDRAULICS.       MATERIALS   OF    ENGINEERING.' 
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Baker's  Engineers'  Surveying  Instruments i amo,  3  oo 

Bixby's  Graphical  Computing  Table Paper  19^X24!  inches.  25 

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Comstock's  Field  Astronomy  for  Engineers 8vo,  2  50 

Davis's  Elevation  and  Stadia  Tables 8vo,  i  oo 

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Fiebeger's  Treatise  on  Civil  Engineering.     (In  press.) 

Folwell's  Sewerage.     (Designing  and  Maintenance.) 8vo,  3  oo 

Freitag's  Architectural  Engineering.     2d  Edition,  Rewritten 8vo,  3  50 

French  and  Ives's  Stereotomy 8vo,  2  50 

Goodhue's  Municipal  Improvements i2mo,  i  75 

Goodrich's  Economic  Disposal  of  Towns'  Refuse 8vo,  3  50 

Gore's  Elements  of  Geodesy 8vo,  2  50 

Hayford's  Text-book  of  Geodetic  Astronomy §vo,  3  oo 

Bering's  Ready  Reference  Tables  (Conversion  Factors) i6mo,  morocco,  2  50 

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Howe's  Retaining  Walls  for  Earth i2mo,  i  25 

Johnson's  (J.  B.)  Theory  and  Practice  of  Surveying Small  8vo,  4  oo 

Johnson's  (L.  J.)  Statics  by  Algebraic  and  Graphic  Methods 8vo,  2  oo 

Laplace's  Philosophical  Essay  on  Probabilities.    (Truscott  and  Emory.) .  i2mo,  2  oo 

Mahan's  Treatise  on  Civil  Engineering.     (1873.)     (Wood.) 8vo,  5  oo 

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Merriman's  Elements  of  Precise  Surveying  and  Geodesy 8vo,  2  50 

Elements  of  Sanitary  Engineering 8vo,  2  oo 

Merriman  and  Brooks's  Handbook  for  Surveyors i6mo,  morocco,  2  oo 

Nugent's  Plane  Surveying 8vo,  3  50 

Ogden's  Sewer  Design i2mo,  2  oo 

Patton's  Treatise  on  Civil  Engineering 8vo  half  leather,  7  50 

Reed's  Topographical  Drawing  and  Sketching 4to,  5  oo 

RideaPs  Sewage  and  the  Bacterial  Purification  of  Sewage 8vo,  3  50 

Siebert  and  Biggin's  Modern  Stone-cutting  and  Masonry „ 8vo,  i  50 

Smith's  Manual  of  Topographical  Drawing.     (McMillan.) 8vo,  2  50 

Sondericker's  Graphic  Statics,  with  Applications  to  Trusses,  Beams,  and  Arches. 

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Taylor  and  Thompson's  Treatise  on  Concrete,  Plain  and  Reinforced 8vo,  5  oo 

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Wait's  Engineering  and  Architectural  Jurisprudence 8vo,  6  oo 

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Foster's  Treatise  on  Wooden  Trestle  Bridges 4to,  5  oo 

Fowler's  Ordinary  Foundations 8vo,  3  50 

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6 


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Church's  Mechanics  of  Engineering 8vo,  6  oo 

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Fowler's  Ordinary  Foundations 8vo,  3  50 

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Merrill's  Stones  for  Building  and  Decoration 8vo,  5  oo 

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Metcalf's  SteeL     A  Manual  for  Steel-users i2mo,  2  oo 

Patton's  Practical  Treatise  on  Foundations _-8vo,  5  oo 

Richardson's  Modern  Asphalt  Pavements 8vo,  3  oo 

Richey's  Handbook  for  Superintendents  of  Construction lomo,  mor.,  4  oo 

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Wood's  (De  V.)  Elements  of  Analytical  Mechanics 8vo,  3  oo 

Wood's  (M.  P.)  Rustless  Coatings:    Corrosion  and  Electrolysis  of  Iron  and 

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Brook's  Handbook  of  Street  Railroad  Location i6mo,  morocco,  i  50 

Butt's  Civil  Engineer's  Field-book i6mo,  morocco,  2  50 

Crandall's  Transition  Curve i6mo,  morocco,  i  50 

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Dawson's  "Engineering"  and  Electric  Traction  Pocket-book.  .  i6mo,  morocco,  5  oo 

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Fisher's  Table  of  Cubic  Yards Cardboard,  25 

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Howard's  Transition  Curve  Field-book i6mo,  morocco,  i  50 

Hudson's  Tables  for  Calculating  the  Cubic  Contents  of  Excavations  and  Em- 
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Molitor  and  Beard's  Manual  for  Resident  Engineers.  . i6mo,  i  oo 

Wagle's  Field  Manual  for  Railroad  Engineers i6mo,  morocco,  3  oo 

Philbrick's  Field  Manual  for  Engineers i6mo,  morocco,  3  oo 

Searles's  Field  Engineering i6mo,  morocco,  3  oo 

Railroad  Spiral i6mo,  morocco,  i  50 

Taylor's  Prismoidal  Formulae  and  Earthwork 8vo,  i  50 

*  Trautwine's  Method  of  Calculating  the  Cube  Contents  of  Excavations  and 

Embankments  by  the  Aid  of  Diagrams 8vo,  2  oo 

The  Field  Practice  of  Laying  Out  Circular  Curves  for  Railroads. 

I2mo,  morocco,  2  50 

Cross-section  Sheet Paper,  25 

Webb's  Railroad  Construction i6mo,  morocco,  5  oo 

Wellington's  Economic  Theory  of  the  Location  of  Railways Small  8vo,  5  oo 

DRAWING. 

Barr's  Kinematics  of  Machinery.- 8vo,  2  50 

*  Bartlett's  Mechanical  Drawing 8vo,  3  oo 

*  "                                         "        Abridged  Ed 8vo,  i  50 

Coolidge's  Manual  of  Drawing 8vo,  paper  i  oo 

Coolidge  and  Freeman's  Elements  of  General  Drafting  for  Mechanical  Engi- 
neers  Oblong  4to,  2  50 

Durley's  Kinematics  of  Machines 8vo,  4  oo 

Bmch's  Introduction  to  Protective  Geometry  and  its  Applications 8vo.  2  50 

8 


Hill's  Text-book  on  Shades  and  Shadows,  and  Perspective 8vo,  2  oo 

Jamison's  Elements  of  Mechanical  Drawing 8vo,  2  50 

Jones's  Machine  Design: 

Part  I.     Kinematics  of  Machinery 8vo,  i  50 

Part  n.     Form,  Strength,  and  Proportions  of  Parts, 8vo,  3  oo 

MacCord's  Elements  of  Descriptive  Geometry 8vo,  3  oo 

Kinematics ;  or,  Practical  Mechanism. 8vo,  5  oo 

Mechanical  Drawing 4to,  4  oo 

Velocity  Diagrams 8vo,  i  50 

*  Mahan's  Descriptive  Geometry  and  Stone-cutting 8vo,  i  50 

Industrial  Drawing.     (Thompson.) 8vo,  3  50 

Moyer's  Descriptive  Geometry 8vo,  2  oo 

Reed's  Topographical  Drawing  and  Sketching 4to,  5  oo 

Reid's  Course  in  Mechanical  Drawing 8vo,  2  oo 

Text-book  of  Mechanical  Drawing  and  Elementary  Machine  Design. 8vo,  3  oo 

Robinson's  Principles  of  Mechanism 8vo,  3  oo 

Schwamb  and  Merrill's  Elements  of  Mechanism 8vo,  3  oo 

Smith's  Manual  of  Topographical  Drawing.     (McMillan.) 8vo,  50 

Warren's  Elements  of  Plane  and  Solid  Free-hand  Geometrical  Drawing.  i2mo,  oo 

Drafting  Instruments  and  Operations i2nm 

Manual  of  Elementary  Projection  Drawing i2mo, 

Manual  of  Elementary*Problems  in  the  Linear  Perspective  of  Form  and 

Shadow i2mo,  oo 

Plane  Problems  in  Elementary  Geometry 12 mo,  25 

Primary  Geometry i2mo,  75 

Elements  K  Descriptive  Geometry,  Shadows,  and  Perspective 8vo,  3  50 

General  Problems  of  Shades  and  Shadows 8vo,  3  oo 

Elements  of  Machine  Construction  and  Drawing :  .8vo,  7  50 

Problems,  Theorems,  and  Examples  in  Descriptive  Geometry 8vo,  2  50 

Weisbach's  Kinematics  and  Power  of  Transmission.    (Hermann  and  Klein)8vo,  5  oo 

Whelpley's  Practical  Instruction  in  the  Ait  of  Letter  Engraving i2mo,  2  oo 

Wilson's  (H.  M.)  Topographic  Surveying 8vo,  3  50 

Wilson's  (V.  T.)  Free-hand  Perspective —  8vo,  2  50 

Wilson's  (V.  T.)  Free-hand  Lettering 8vo,  i  oo 

Woo  If 's  Elementary  Course  in  Descriptive  Geometry Large  8vo,  3  oo 


ELECTRICITY  AND  PHYSICS. 

Anthony  and  Brackett's  Text-book  of  Physics.     (Magie.) Small  8vo,  3  oo 

Anthony's  Lecture-notes  on  the  Theory  of  Electrical  Measurements. . . .  12010,  i  oo 

Benjamin's  History  of  Electricity 8vo,  3  oo 

Voltaic  CelL ; 8vo,  3  oo 

Classen's  Quantitative  Chemical  Analysis  by  Electrolysis.     (Boltwood.).8vo,  3  oo 

Crehore  and  Squier's  Polarizing  Photo-chronograph 8vo,  3  oo 

Dawson's  "Engineering"  and  Electric  Traction  Pocket-book.  i6mo,  morocco,  5  oo 
Dolezalek's   Theory   of   the    Lead   Accumulator    (Storage    Battery).      (Von 

Ende.) I2mo,  2  50 

Duhem's  Thermodynamics  and  Chemistry.     (Burgess.) 8vo,  4  oo 

Flather's  Dynamometers,  and  the  Measurement  of  Power i2mo,  3  oo 

Gilbert's  De  Magnete.     (Mottelay.) 8vo,  2  50 

Hanchett's  Alternating  Currents  Explained I2mo,  i  oo 

Bering's  Ready  Reference  Tables  (Conversion  Factors) i6mo,  morocco,  2  50 

Holman's  Precision  of  Measurements • 8vo,  2  oo 

Telescopic   Mirror-scale  Method,  Adjustments,  and  Tests Large  8vo,  75 

Kinzbrunner's  Testing  of  Continuous-Current  Machines. 8vo,  2  oo 

Landauer's  Spectrum  Analysis.     (Tingle.) 8vo,  3  oo 

Le  Chatelien's  High-temperature  Measurements.  (Boudouard — Burgess.)  i2mo,  3  oo 

Lob's  Electrolysis  and  Electrosynthesis  of  Organic  Compounds.  (Lorenz.)  i2mo,  i  oo 

9 


•*  Lyons's  Treatise  on  Electromagnetic  Phenomena.  Vols.  I.  and  II.  8vo,  each,    6  oo 

*  Michie's  Elements  of  Wave  Motion  Relating  to  Sound  and  Light 8vo,    4  oo 

Niaudet's  Elementary  Treatise  on  Electric  Batteries.     (Fishback.) i2mo, 


*  Rosenberg's  Electrical  Engineering.     (Haldane  Gee — Kinzbrunner.).  .  .8vo, 

Ryan,  Norris,  and  Hoxie's  Electrical  Machinery.     Vol.  1 8vo, 

Thurston's  Stationary  Steam-engines 8vo, 

*  Tillman's  Elementary  Lessons  in  Heat 8vo, 

Tory  and  Pitcher's  Manual  of  Laboratory  Physics Small  8vo,  2  oo 

Hike's  Modern  Electrolytic  Copper  Refining „ 8vo,  3  oo 

LAW. 

*  Davis's  Elements  of  Law 8vo,  2  50 

*  Treatise  on  the  Military  Law  of  United  States 8vo,  7  oo 

*  Sheep,  7  50 

Manual  for  Courts-martial i6mo,  morocco,  i  50 

Wait's  Engineering  and  Architectural  Jurisprudence 8vo,  6  oo 

Sheep,  6  50 

Law  of  Operations  Preliminary  to  Construction  in  Engineering  and  Archi- 
tecture  8vo,  5  oo 

Sheep,  5  50 

Law  of  Contracts 8vo,  3  oo 

Winthrop's  Abridgment  of  Military  Law I2mo,  2  50 

MANUFACTURES. 

Bernadou's  Smokeless  Powder — Nitro-cellulose  and  Theory  of  the  Cellulose 

Molecule i2mo,  2  50 

Bolland's  Iron  Founder i2mo,  2  50 

"The  Iron  Founder,"  Supplement i2mo,  2  50 

Encyclopedia  of  Founding  and  Dictionary  of  Foundry  Terms  Used  in  the 

Practice  of  Moulding i2mo,  3  oo 

Eissler's  Modern  High  Explosives 8vo,  4  oo 

Effront's  Enzymes  and  their  Applications.     (Prescott.) 8vo,  3  oo 

Fitzgerald's  Boston  Machinist i2mo,  i  oo 

Ford's  Boiler  Making  for  Boiler  Makers i8mo,  i  oo 

Hopkin's  Oil-chemists'  Handbook 8vo,  3  oo 

Keep's  Cast  Iron 8vo,  2  50 

Leach's  The  Inspection  and  Analysis  of  Food  with  Special  Reference  to  State 

Control Large  8vo,  7  50 

Matthews's  The  Textile  Fibres 8vo,  3  50 

Metcalf 's  Steel.     A  Manual  for  Steel-users i2mo,  2  oo 

Metcalfe's  Cost  of  Manufactures — And  the  Administration  of  Workshops. 8vo,  5  oo 

Meyer's  Modern  Locomotive  Construction 4to,  10  oo 

Morse's  Calculations  used  in  Cane-sugar  Factories i6mo,  morocco,  i  50 

*  Reisig's  Guide  to  Piece-dyeing 8vo,  25  oo 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish 8vo,  3  oo 

Smith's  Press-working  of  Metals 8vo,  3  oo 

Spalding's  Hydraulic  Cement i2mo,  2  oo 

Spencer's  Handbook  for  Chemists  of  Beet-sugar  Houses.    ...  i6mo,  morocco,  3  oo 

Handbook  for  Sugar  Manufacturers  and  their  Chemists.  .  i6mo,  morocco,  ^  oo 

Taylor  and  Thompson's  Treatise  on  Concrete,  Plain  and  Reinforced 8vo,  5  oo 

Thurston's  Manual  of  Steam-boilers,  their  Designs,  Construction  and  Opera- 
tion  8vo,  5  oo 

*  Walke's  Lectures  on  Explosives 8vo,  4  oo 

Ware's  Manufacture  of  Sugar.     (In  press.) 

West's  American  Foundry  Practice i2mo,  2  50 

Moulder's  Text-book i2mo,  2  50 

10 


Wolff's  Windmill  as  a  Prime  Mover 8vo,    3  oo 

Wood's  Rustless  Coatings:   Corrosion  and  Electrolysis  of  Iron  and  Steel.  .8vo,    4  oo 


MATHEMATICS. 

Baker's  Elliptic  Functions 8vo,  g« 

*  Bass's  Elements  of  Differential  Calculus -. i2mo,  oo 

Briggs's  Elements  of  Plane  Analytic  Geometry 12 mo,  »oo 

Compton's  Manual  of  Logarithmic  Computations 12 mo,  50 

Davis's  Introduction  to  the  Logic  of  Algebra 8vo,  50 

*  Dickson's  College  Algebra Large  i2mo,  50 

*  Introduction  to  the  Theory  of  Algebraic  Equations Large  i2mo,  25 

Emch's  Introduction  to  Protective  Geometry  and  its  Applications 8vo,  50 

Halsted's  Elements  of  Geometry 8vo,  73 

Elementary  Synthetic  Geometry 8vo,  50 

Rational  Geometry i amo,  75 

*,,Johnson's  (J.  B.)  Three-place  Logarithmic  Tables:   Vest-pocket  size. paper,  15 

100  copies  for  5  oo 

*  Mounted  on  heavy  cardboard,  8X  10  inches,  25 

10  copies  for  2  oo 

Johnson's  (W.  W.)  Elementary  Treatise  on  Differential  Calculus .  .  Small  8vo,  ^  oo 

Johnson's  (W.  W.)  Elementary  Treatise  on  the  Integral  Calculus. Small  8vo,  i  50 

Johnson's  (W.  W.)  Curve  Tracing  in  Cartesian  Co-ordinates i2mo,  i  oo 

Johnson's  (W.  W.)  Treatise  on  Ordinary  and  Partial  Differential  Equations. 

Small  8vo,  3  50 

Johnson's  (W.  W.)  Theory  of  Errors  and  the  Method  of  Least  Squares.  i2mo,  i  50 

*  Johnson's  (W.  W.)  Theoretical  Mechanics i2mo,  3  oo 

Laplace's  Philosophical  Essay  on  Probabilities.     (Truscott  and  Emory.) .  i2mo,  2  oo 

*  Ludlow  and  Bass.     Elements  of  Trigonometry  and  Logarithmic  and  Other 

Tables .  .8vo,  3  oo 

Trigonometry  and  Tables  published  separately Each,  2  oo 

*  Ludlow's  Logarithmic  and  Trigonometric  Tables 8vo,  i  oo 

Maurer's  Technical  Mechanics &» . ,  4  oo 

Merriman  and  Woodward's  Higher  Mathematics. 8vo,  5  oo 

Merriman's  Method  of  Least  Squares 8vo,  2  oo 

Rice  and  Johnson's  Elementary  Treatise  on  the  Differential  Calculus. .  Sm.  8vo,  3  oo 

Differential  and  Integral  Calculus.     2  vols.  in  one Small  8vo,  2  50 

Wood's  Elements  of  Co-ordinate  Geometry 8vo,  2  oo 

Trigonometry:  Analytical,  Plane,  and  Spherical , i2mo,  i  oo 


MECHANICAL  ENGINEERING. 

MATERIALS  OF  ENGINEERING,  STEAM-ENGINES  AND  BOILERS. 

Bacon's  Forge  Practice i2mo,  i  50 

Baldwin's  Steam  Heating  for  Buildings i2mo,  2  50 

Barr's  Kinematics  of  Machinery 8vo,  2  50 

*  Bartlett's  Mechanical  Drawing 8vo,  3  oo 

*  "  "  "        Abridged  Ed 8vo,     150 

Benjamin's  Wrinkles  and  Recipes i2mo,    2  oo 

Carpenter's  Experimental  Engineering 8vo,    6  oo 

Heating  and  Ventilating  Buildings 8vo,    4  oo 

Cary's  Smoke  Suppression  in  Plants  using  Bituminous  CoaL     (In  Prepara- 
tion.) 

Clerk's  Gas  and  Oil  Engine Small  8vo,    4  oo 

Coolidge's  Manual  of  Drawing 8vo,  paper,     i  oo 

Coolidge  and  Freeman's  Elements  of  General  Drafting  for  Mechanical  En- 
gineers  , , , , , Oblong  4to,    2  50 

11 


Cromwell's  Treatise  on  Toothed  Gearing .* tamo,  i  50 

Treatise  on  Belts  and  Pulleys I2mo,  i  50 

Durley's  Kinematics  of  Machines 8vo,  4  oo 

Flather's  Dynamometers  and  the  Measurement  of  Power i2mo,  3  oo 

^  Rope  Driving i2mo,  2  oo 

Gill's  Gas  and  Fuel  Analysis  for  Engineers i2mo,  i  25 

Hall's  Car  Lubrication I2mo,  i  oo 

Bering's  Ready  Reference  Tables  (Conversion  Factors) i6mo,  morocco,  2  50 

Button's  The  Gas  Engine 8vo,  5  oo 

Jamison's  Mechanical  Drawing 8vo,  2  50 

Jones's  Machine  Design: 

Part  I.     Kinematics  of  Machinery 8vo,  i  50 

Part  n.     Form,  Strength,  and  Proportions  of  Parts 8vo,  3  oo 

Kent's  Mechanical  Engineers'  Pocket-book i6mo,  morocco,  5  oo 

Kerr's  Power  and  Power  Transmission 8vo,  2  oo 

Leonard's  Machine  Shop,  Tools,  and  Methods.     (In  press.) 

Lorenz's  Modern  Refrigerating  Machinery.     (Pope,  Haven,  and  Dean.)     (In  press.) 

MacCord's  Kinematics;   or,  Practical  Mechanism 8vo,  5  oo 

Mechanical  Drawing 4to,  4  oo 

Velocity  Diagrams 8vo,  i  50 

Mahan's  Industrial  Drawing.     (Thompson.) 8vo,  3  50 

Poole*s  Calorific  Power  of  Fuels 8vo,  3  oo 

Reid's  Course  in  Mechanical  Drawing 8vo,  2  oo 

Text-book  of  Mechanical  Drawing  and  Elementary  Machine  Design. 8vo,  3  oo 

Richard's  Compressed  Air i2mo,  i  50 

Robinson's  Principles  of  Mechanism 8vo,  3  oo 

Schwamb  and  Merrill's  Elements  of  Mechanism 8vo,  3  oo 

Smith's  Press-working  of  Metals 8vo,  3  oo 

Thurston's   Treatise   on   Friction  and   Lost   Work   in   Machinery   and   Mill 

Work 8vo,  3  oo 

Animal  as  a  Machine  and  Prime  Motor,  and  the  Laws  of  Energetics .  i2mo,  i  oo 

Warren's  Elements  of  Machine  Construction  and  Drawing ,  .  .  .8vo,  7  50 

Weisbach's    Kinematics    and    the    Power    of    Transmission.     (Herrmann — 

Klein.) 8vo,  5  oo 

Machinery  of  Transmission  and  Governors.     (Herrmann — Klein.).  .8vo,  5  oo 

Wolff's  Windmill  as  a  Prime  Mover 8vo,  3  oo 

Wood's  Turbines 8vo,  2  50 


MATERIALS   OF   ENGINEERING. 

Bovey's  Strength  of  Materials  and  Theory  of  Structures 8vo,  7  50 

Burr's  Elasticity  and  Resistance  of  the  Materials  of  Engineering.    6th  Edition. 

Reset 8vo,  7  50 

Church's  Mechanics  of  Engineering 8vo,  6  oo 

Johnson's  Materials  of  Construction 8vo,  6  oo 

Keep's  Cast  Iron 8vo,  2  50 

Lanza's  Applied  Mechanics , 8vo,  7  50 

Martens's  Handbook  on  Testing  Materials.     (Henning.) 8vo,  7  50 

Merriman's  Text-book  on  the  Mechanics  of  Materials 8vo,  4  oo 

Strength  of  Materials i2mo,  i  oo 

Metcalf 's  Steel.     A  manual  for  Steel-users i2mo.  2  oo 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish 8vo,  3  oo 

Smith's  Materials  of  Machines I2mo,  i  oo 

Thurston's  Materials  of  Engineering 3  vols.,  8vo,  8  oo 

Part  II.     Iron  and  Steel 8vo,  3  50 

Part  IH.     A  Treatise  on  Brasses,  Bronzes,  and  Other  Alloys  and  their 

Constituents 8vo,  2  50 

Text-book  of  the  Materials  of  Construction 8vo,  5  <>• 

12 


Wood's  (De  V.)  Treatise  on  the  Resistance  of  Materials  and  an  Appendix  on 

the  Preservation  of  Timber 8vo,    2  oe 

Wood's  (De  V.)  Elements  of  Analytical  Mechanics. 8vo,    3  oo 

food's  (M.  P.)  Rustless  Coatings:   Corrosion  and  Electrolysis  of  Iron  and 

SteeL 8vo,    4  oo 


STEAM-ENGINES  AND  BOILERS. 

Berry's  Temperature-entropy  Diagram. I2mo,  i  23 

Carnot's  Reflections  on  the  Motive  Power  of  Heat     (Thurston.) i2mo,  i  50 

Dawson's  "Engineering"  and  Electric  Traction  Pocket-book. . .  .i6mo,  mor.,  5  oo 

Ford's  Boiler  Making  for  Boiler  Makers i8mo,  i  oo 

Goss's  Locomotive  Sparks 8vo,  2  oo 

Hemenway's  Indicator  Practice  and  Steam-engine  Economy 12 mo,  2  oo 

Button's  Mechanical  Engineering  of  Power  Plants 8vo,  5  oo 

Heat  and  Heat-engines 8vo,  5  oo 

Kent's  Steam  boiler  Economy 8vo,  4  oo 

Kneass's  Practice  and  Theory  of  the  Injector 8vo,  i  50 

MacCord's  Slide-valves 8vo,  2  oo 

Meyer's  Modern  Locomotive  Construction 4to,  10  oo 

Peabody's  Manual  of  the  Steam-engine  Indicator 12 mo.  i  50 

Tables  of  the  Properties  of  Saturated  Steam  and  Other  Vapors 8vo,  i  oo 

Thermodynamics  of  the  Steam-engine  and  Other  Heat-engines 8vo,  5  oo 

Valve-gears  for  Steam-engines. 8vo,  2  50 

Peabody  and  Miller's  Steam-boilers ; .  .  .  .8vo,  4  oo 

Pray's  Twenty  Years  with  the  Indicator Large  8vo,  2  50 

Pupin's  Thermodynamics  of  Reversible  Cycles  in  Gases  and  Saturated  Vapors. 

(Osterberg.) i2mo,  i  25 

Reagan's  Locomotives:  Simple   Compound,  and  Electric i2mo,  2  50 

Rontgen's  Principles  of  Thermodynamics.     (Du  Bois.) 8vo,  5  oo 

Sinclair's  Locomotive  Engine  Running  and  Management 12 mo,  2  oo 

Smart's  Handbook  of  Engineering  Laboratory  Practice i2mo,  2  50 

%iow's  Steam-boiler  Practice 8vo,  3  oo 

6pangler's  Valve-gears 8vo,  2  50 

Notes  on  Thermodynamics ." i2mo,  i  oo 

Spangler,  Greene,  and  Marshall's  Elements  of  Steam-engineering 8vo,  3  oo 

Thurston's  Handy  Tables 8vo.  i  50 

Manual  of  the  Steam-engine 2  vols.,  8vo,  10  oo 

Part  I.     History,  Structure,  and  Theory 8vo,  6  oo 

Part  n.     Design,  Construction,  and  Operation 8vo,  6  oo 

Handbook  of  Engine  and  Boiler  Trials,  and  the  Use  of  the  Indicator  and 

the  Prony  Brake 8vo,  5  oo 

Stationary  Steam-engines 8vo,  2  50 

Steam-boiler  Explosions  in  Theory  and  in  Practice 1 2mo ,  i  50 

Manual  of  Steam-boilers,  their  Designs,  Construction,  and  Operation 8vo,  5  oo 

Weisbach's  Heat.  Steam,  and  Steam-engines.     (Du  Bois.) 8vo,  5  oo 

Whitham's  Steam-engine  Design 8vo,  5  oo 

Wilson's  Treatise  on  Steam-boilers.     (Flather.) i6mo,  a  50 

Wood's  Thermodynamics,  Heat  Motors,  and  Refrigerating  Machines. .  .8vo,  4  oo 


MECHANICS  AND  MACHINERY. 

Barr's  Kinematics  of  Machinery 8vo,  2  50 

Bovey's  Strength  of  Materials  and  Theory  of  Structures 8vo,  7  50 

Chase's  The  Art  of  Pattern-making I2mo,  2  50 

Church's  Mechanics  of  Engineering 8vo,  6  oo 

13 


Church's  Notes  and  Examples  in  Mechanics 8vo,  2  oo 

Compton's  First  Lessons  in  Metal-working izmo,  i  50 

Compton  and  De  Groodt's  The  Speed  Lathe i2mo,  i  50 

Cromwell's  Treatise  on  Toothed  Gearing i2mo,  i  50 

Treatise  on  Belts  and  Pulleys i2mo,  i  50 

Dana's  Text-book  of  Elementary  Mechanics  for  Colleges  and  Schools.  .  i2mo,  i  50 

Dingey's  Machinery  Pattern  Making i2mo,  2  oo 

Dredge's  Record  of  the  Transportation  Exhibits  Building  of  the  World's 

Columbian  Exposition  of  1893 4to  half  morocco,  5  oo 

Du  Bois's  Elementary  Principles  of  Mechanics: 

Vol.      I.     Kinematics 8vo,  3  50 

Vol.    II.     Statics 8vo,  400 

VoL  in.     Kinetics , 8vo,  3  50 

Mechanics  of  Engineering.     Vol.    I Small  4to,  7  So 

VoL  II Small  4to,  10  oo 

Durley's  Kinematics  of  Machines 8vo,  4  oo 

Fitzgerald's  Boston  Machinist i6mo,  i  oo 

Flather's  Dynamometers,  and  the  Measurement  of  Power i2mo,  3  oo 

Rope  Driving i2mo,  2  oo 

Goss's  Locomotive  Sparks 8vo,  2  oo 

Hall's  Car  Lubrication i2mo,  i  oo 

Holly's  Art  of  Saw  Filing i8mo,  75 

James's  Kinematics  of  a  Point  and  the  Rational  Mechanics  of  a  Particle.  Sm.8vo,2  oo 

*  Johnson's  (W.  W.)  Theoretical  Mechanics i2mo,  3  oo 

Johnson's  (L.  J.)  Statics  by  Graphic  and  Algebraic  Methods 8vo,  2  oo 

Jones's  Machine  Design: 

Part    I.     Kinematics  of  Machinery 8vo,  i  50 

Part  II.     Form,  Strength,  and  Proportions  of  Parts 8vo,  3  oo 

Kerr's  Power  and  Power  Transmission 8vo,  2  oo 

Lanza's  Applied  Mechanics. 8vo,  7  50 

Leonard's  Machine  Shop,  Tools, -and  Methods.     (In  press.) 

Lorenz's  Modern  Refrigerating  Machinery.      (Pope,  Haven,  and  Dean.)      (In  press.) 

MacCord's  Kinematics;  or,  Practical  Mechanism 8vo,  5  oo 

Velocity  Diagrams 8vo,  i  50 

Maurer's  Technical  Mechanics 8vo,  4  oo 

Merriman's  Text-book  on  the  Mechanics  of  Materials 8vo,  4  oo 

*  Elements  of  Mechanics i2mo,  i  oo 

*  Michie's  Elements  of  Analytical  Mechanics 8vo,  4  oo 

Reagan's  Locomotives:  Simple,  Compound,  and  Electric i2mo>  2  50 

Reid's  Course  in  Mechanical  Drawing 8vo,  2  oo 

Text-book  of  Mechanical  Drawing  and  Elementary  Machine  Design.  8vo,  3  oo 

Richards's  Compressed  Air i2mo,  i  50 

Robinson's  Principles  of  Mechanism 8vo,  3  oo 

Ryan,  Norris,  and  Hoxie's  Electrical  Machinery.     Vol.  1 8vo,  2  50 

Schwamb  and  Merrill's  Elements  of  Mechanism 8vo,  3  oo 

Sinclair's  Locomotive-engine  Running  and  Management i2mo,  2  oo 

Smith's  (O.)  Press-working  of  Metals. 8vo,  3  oo 

Smith's  (A.  W.)  Materials  of  Machines I2mo,  i  oo 

Spangler,  Greene,  and  Marshall's  Elements  of  Steam-engineering 8vo,  3  oo 

Thurston's  Treatise  on  Friction  and  Lost  Work  in    Machinery  and    Mill 

Work 8vo,  3  oo 

Animal  as  a  Machine  and  Prime  Motor,  and  the  Laws  of  Energetics. 

i2mo,  i  oo 

Warren's  Elements  of  Machine  Construction  and  Drawing 8vo,  7  50 

Weisbach's  Kinematics  and  Power  of  Transmission.   ( Herrmann — Klein. ) .  8vo ,  5  oo 

Machinery  of  Transmission  and  Governors.      (Herrmann — Klein. ).8vo,  5  oo 

Wood's  Elements  of  Analytical  Mechanics 8vo,  3  oo 

Principles  of  Elementary  Mechanics I2mo,  i  25 

Turbines 8vo,  2  50 

The  World's  Columbian  Exposition  of  1893 4to,  i  oo 

14 


METALLURGY. 

£gleston's  Metallurgy  of  Silver,  Gold,  and  Mercury: 

Vol.    L     Silver 8vo,  7  5» 

VoL  n.     Gold  and  Mercury 8vo,  7  50 

**  Iles's  Lead-smelting.     (Postage  9  cents  additional.) I2mo,  2  50 

Keep's  Cast  Iron 8vo,  2  50 

Kunhardt's  Practice  of  Ore  Dressing  in  Europe .8vo,  i  go 

Le  Chatelier's  High-temperature  Measurements.  (Boudouard — Burgess.  )i2mo,  3  oo 

Metcalf's  Steel     A  Manual  for  Steel-users-     i2mo,  2  oo 

Smith's  Materials  of  Machines I2mo,  i  oo 

Thurston's  Materials  of  Engineering.     In  Three  Parts 8vo,  8  oo 

Part    n.     Iron  and  SteeL  .  . . .' 8vo.  3  50 

Part  HI.     A  Treatise  on  Brasses,  Bronzes,  and  Other  Alloys  and  their 

Constituents 8vo,  2  50- 

Ulke's  Modern  Electrolytic  Copper  Refining 8vo,  3  oo 

MINERALOGY. 

Barringer's  Description  of  Minerals  of  Commercial  Value.    Oblong,  morocco,  2  50 

Boyd's  Resources  of  Southwest  Virginia 8vo  3  oo 

Map  of  Southwest  Virignia Pocket-book  form.  2  oo 

Brush's  Manual  of  Determinative  Mineralogy.     (Penfield.) 8vo,  4  oo 

Chester's  Catalogue  of  Minerals 8vo,  paper,  i  oo 

»                                                               Cloth,  i  25 

Dictionary  of  the  Names  of  Minerals 8vo,  3  50 

Dana's  System  of  Mineralogy Large  8vo,  half  leather,  12  50 

First  Appendix  to  Dana's  New  "  System  of  Mineralogy." Large  8vo,  i  oo 

Text-book  of  Mineralogy 8vo,  4  oo 

Minerals  and  How  to  Study  Them I2mo,  i  so 

Catalogue  of  American  Localities  of  Minerals Large  8vo,  i  oo 

Manual  of  Mineralogy  and  Petrography i2mo,  2  oo 

Douglas's  Untechnical  Addresses  on  Technical  Subjects 12 mo,  i  oo 

Eakle's  Mineral  Tables 8vo,  i  25 

Egleston's  Catalogue  of  Minerals  and  Synonyms 8vo,  2  50 

Hussak's  The  Determination  of  Rock-forming  Minerals.    (Smith.). Small  8vo,  2  oo 

Merrill's  Non-metallic  Minerals:   Their  Occurrence  and  Uses 8vo.  4  oo 

*  Pbnfieid's  Notes  on  Determinative  Mineralogy  and  Record  of  Mineral  Tests. 

8vo.  paper,  o  50 
Roseabusch's   Microscopical   Physiography   of   the   Rock-making  Minerals. 

(Iddings.) 8vo.  5  oo 

*  Tillman's  Text-book  of  Important  Minerals  and  Rocks 8vo.  2  oo 

Williwns's  Manual  of  Lithology 8vo,  3  oo 

MINING. 

fceard's  Ventilation  of  Mines I2mo.  2  50 

Boyd's  Resources  of  Southwest  Virginia 8vo.  3  oo 

Mep  of  Southwest  Virginia Pocket  book  form,  2  oo 

Douglac's  Untechnical  Addresses  on  Technical  Subjects i2mo.  i  oo 

*  Drinker's  Tunneling,  Explosive  Compounds,  and  Rock  Drills.  .4to,hf.  mor..  25  oo 

Eissler's  Modern  High  Explosives 8vo.  4  oo 

Fowler's  Sewage  Works  Analyses i2tno  2  oo 

Goodyear 's  Coal-mines  of  the  Western  Coast  of  the  United  States 12 mo.  2  50 

Ihlseng's  Manual  of  Mining 8vo.  5  oo 

**  Iles's  Lead-smelting.     (Postage  QC.  additional ) I2mo.  2  50 

Kunhardt's  Practice  of  Ore  Dressing  in  Europe 8vo.  i  50 

O'Driscoll's  Notes  on  the  Treatment  of  Gold  Ores 8vo.  2  oo 

*  Walke's  Lectures  on  Explosives 8vo,  4  oo 

Wilson's  Cyanide  Processes I2mo,  i  50 

Chlosination  Process. iamo,  j  5O 

15 


Wilson's  HydrauLw  and  Placer  Mining i2mo,  2  oo 

Treatise  on  Practical  and  Theoretical  Mine  Ventilation T2mo,  i  25 

SANITARY  SCIENCE. 

Folwell's  Sewerage.     (Designing,  Construction,  and  Maintenance.) 8vo,  3  oc 

Water-supply  Engineering 8vo,  4  oo 

Fuertes's  Water  and  Public  Health i2mo,  i  50 

Water-filtration  Works I2mo,  2  50 

Gerhard's  Guide  to  Sanitary  House-inspection i6mo,  i  oo 

Goodrich's  Economic  Disposal  of  Town's  Refuse Demy  8vo,  3  50 

Hazen's  Filtration  of  Public  Water-supplies 8vo,  3  oo 

Leach's  The  Inspection  and  Analysis  of  Food  with  Special  Reference  to  State 

Control 8vo,  7  50 

Mason's  Water-supply.  (Considered  principally  from  a  Sanitary  Standpoint)  8vo,  4  oo 

Examination  of  Water.     (Chemical  and  Bacteriological.) i2mo,  i  25 

Merriman's  Elements  of  Sanitary  Engineering 8vo,  2  oo 

Ogden's  Sewer  Design 1 2mo,  2  oo 

Prescott  and  Winslow's  Elements  of  Water  Bacteriology,  with  Special  Refer- 
ence to  Sanitary  Water  Analysis I2mo,  25 

*  Price's  Handbook  on  Sanitation I2mo,  50 

Richards's  Cdst  of  Food.     A  Study  in  Dietaries I2mo,  oo 

Cost  of  Living  as  Modified  by  Sanitaiy  Science i2mo,  oo 

Richards  and  Woodman's  Air,  Water,  and  Food  from  a  Sanitary  Stand- 
point     * 8vo,  oo 

*  Richards  and  Williams's  The  Dietary  Computer 8vo,  50 

Rideal's  Sewage  and  Bacterial  Purification  of  Sewage 8vo,  3  50 

Turneaure  and  Russell's  Public  Water-supplies 8vo,  5  oo 

Von  Behring's  Suppression  of  Tuberculosis.     (Bolduan.) I2mo,  i  oo 

Whipple's  Microscopy  of  Drinking-water 8vo,  3  50 

Woodhull's  Notes  on  Military  Hygiene i6mo,  i  50 

MISCELLANEOUS. 

De  Fursac's  Manual  of  Psychiatry.     (Rosanoff  and  Collins.).  ..  .Large  i2mo,  2  50 
Emmons's  Geological  Guide-book  of  the  Rocky  Mountain  Excursion  of  the 

International  Congress  of  Geologists Large  8vo,  i  50 

Fen-el's  Popular  Treatise  on  the  Winds 8vo.  4  oo 

Haines's  American  Railway  Management i2mo,  2  50 

Mott's  Composition,  Digestibility,  and  Nutritive  Value  of  Food.  Mounted  chart,  i  25 

Fallacy  of  the  Present  Theory  of  Sound i6mo,  i  oo 

Ricketts's  History  of  Rensselaer  Polytechnic  Institute,  1824-1894.  .Small  8vo,  3  oo 

Rostoski's  Serum  Diagnosis.     (Bolduan.) i2mo,  i  oo 

Rother ham's  Emphasized  New  Testament Large  8vo,  2  oo 

Steel's  Treatise  on  the  Diseases  of  the  Dog 8vo,  3  so 

Totten's  Important  Question  in  Metrology 8vo,  2  50 

The  World's  Columbian  Exposition  of  1893 4to,  i  oo 

Von  Behring's  Suppression  of  Tuberculosis.     (Bolduan.) i2mo,  i  oo 

Winslow's  Elements  of  Applied  Microscopy i2mo,  i  50 

Worcester  and  Atkinson.     Small  Hospitals,  Establishment  and  Maintenance; 

Suggestions  for  Hospital  Architecture :  Plans  for  Small  Hospital .  12 mo,  123 

HEBREW  AND  CHALDEE  TEXT-BOOKS. 

Green's  Elementary  Hebrew  Grammar i2mo,  i  25 

Hebrew  Chrestomathy 8vo,  2  oo 

Gesenius's  Hebrew  and  Chaldee  Lexicon  to  the  Old  Testament  Scriptures. 

(Tregelles.) Small  4to,  half  morocco,  5  oo 

Lettesis's  Hebrew  Bible 8vo»  2  25 


THIS  BOOK 


ov£RDUE.          «..oo   ON  THE  N 


